day 4 sept 12 lesson 1-3 laws of exponents.notebook
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Day 4 sept 12 lesson 1-3 Laws of
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Write the expression using exponents. 1. (–5)(–5)(–5)(–5)
SOLUTION: The base –5 is a factor 4 times, so the exponent is 4.
(–5)(–5)(–5)(–5) = (–5) 4
2. 3 • 3 • 5 • q • q • q
SOLUTION: The base 3 is a factor 2 times, so its exponent is 2. The base 5 is a factor 1 time, so its exponent is 1. Because a number without an exponent is the same as a number with an exponent of 1, write 5 without an exponent. The base q is a factor 3 times, so its exponent is 3.
3 • 3 • 5 • q • q • q = 3 2 • 5 • q3
3. m • m • m • m • m
SOLUTION: The base m is a factor 5 times, so its exponent is 5.
m • m • m • m • m = m 5
Evaluate the expression.
4. (–9) 4
(–9) 4 = (–9) • (–9) • (–9) • (–9) or 6,561
5.
or
6.
7. In the United States, nearly 8 • 10 9 text messages
are sent every month. About how many text messages is this?
SOLUTION: Write the power as a product and multiply.
There are about 8 billion text messages sent in the United States each month.
8. Interstate 70 stretches almost 2 3 • 5
2 • 11 miles
across the United States. About how many miles long is Interstate 70?
SOLUTION: Write the power as a product and multiply.
Interstate 70 stretches for almost 2,200 miles.
Evaluate the expression.
9. g5 – h 3 if g = 2 and h = 7
SOLUTION: Replace g with 2 and h with 7 in the expression
. Then write the powers as products and simplify.
10. c 2 + d
3 if c = 8 and d = –3
SOLUTION: Replace c with 8 and d with –3 in the expression
. Then write the powers as products and simplify.
11. a 2 • b
SOLUTION:
Replace a with and b with 2 in the expression
. Then write the powers as products and simplify.
12. (r – s) 3 + r
2 if r = –3 and s = –4
SOLUTION: Replace r with –3 and s with –4 in the expression
. Then write the powers as products and simplify.
13. Model with Mathematics Refer to the graphic novel frame below for Exercises a–c.
The metric system is based on powers of 10. For example, one kilometer is equal to 1,000 meters or
10 3 meters. Write each measurement in meters as a
power of 10. a. megameter (1,000,000 meters) b. gigameter (1,000,000,000 meters) c. petameter (1,000,000,000,000,000 meters)
SOLUTION: a. megameter: 1,000,000 meters has 6 zeros or
powers of 10. So, a megameter is 10 6 .
b. gigameter: 1,000,000,000 meters has 9 zeros or
powers of 10. So, a gigameter is 10 9 .
c. petameter: 1,000,000,000,000,000 meters has 15
.
14. Identify Structure Write an expression with an exponent that has a value between 0 and 1.
SOLUTION:
3 = 27, 3
2 = 9, 3
1 = 3.
.
1 = 3
To go from 81 to 27, divide 81 by 3. This is the same for the rest of the values: 27 ÷ 3 = 9, and 9 ÷ 3 = 3.
To find the value of 2 -1
, follow the same pattern.
2 3 = 8, 2
0 = 2 ÷ 2 or
2 3 • 3
4
A. 3 • 3 • 4 • 4 • 4 B. 2 • 2 • 2 • 3 • 3 • 3 C. 2 • 2 • 2 • 3 • 3 • 3 • 3 D. 6 • 12
SOLUTION:
2 3 • 34
= 2 • 2 • 2 • 3 • 3 • 3 • 3. This corresponds to choice C.
17. Write 3 · p p p · 3 · 3 using exponents.
18. Evaluate x3 + y 4 if x = −3 and y = 4.
SOLUTION: Replace x with –3 and y with 4 in the expression
. Then write the powers as products and simplify.
Write the expression using exponents.
19.
SOLUTION:
The base is a factor 3 times, so its exponent is
3.
20. s • (–7) • s • (–7) • (–7)
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base –7 is a factor 3 times, so its exponent is 3. The base s is a factor 2 times, so its exponent is 2.
21. 4 • b • b • 4 • b • b
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base 4 is a factor 2 times, so its exponent is 2. The base b is a factor 4 times, so its exponent is 4.
Evaluate the expression.
SOLUTION:
Replace k with 3 and m with in the expression
k 4 • m. Then write the powers as products and
simplify.
3 , if c = –1 and d = 2
SOLUTION: Replace c with –1 and d with 2 in the expression
. Then write the powers as
products and simplify.
statement.
2
SOLUTION: Simplify each expression.
28 is larger than 25, so > will make the statement true.
(6 – 2) 2 + 3 • 4 > 5
2
3 3
SOLUTION: Simplify each expression.
81 is equal to 81, so = will make the statement true.
5 + 7 2 + 3
true.
=
27. Multiple Representations A square has a side length of s inches. a. Tables Copy and complete the table showing the side length, perimeter, and area of the square on a separate piece of paper.
b. Graphs On a separate piece of grid paper, graph the ordered pairs (side length, perimeter) and (side length, area) on the same coordinate plane. Then connect the points for each set. c. Words On a separate piece of paper, compare and contrast the graphs of the perimeter and area of the square. Which graph is a line?
SOLUTION: a. The perimeter of a square is found using the expression 4s. The area of a square is found using
the expression s 2 . Complete the table.
b. Graph the values from the table on the same coordinate plane. Side length is represented on the x- axis and both perimeter and area are represented on the y-axis.
c. Sample answer: The graph representing perimeter of a square is a line because each side length is multiplied by 4. The graph representing area of a square is nonlinear because each side length is squared. Squared values do not increase at a constant rate.
28. To find the volume of a cube, multiply its base, its height, and its width.
What is the volume of the cube expressed as a power?
A. 6 2 in
3
SOLUTION: A cube has the same measures for all sides, so the base, height, and width are all 6 inches. The volume
is 6 • 6 • 6 or 6 3 inches cubed. This corresponds to
choice B.
29. Short Response The volume of an ice cube in
cubic millimeters is represented by the term 11 3 .
What is 11 3 in standard form?
SOLUTION:
30. What is the value of x2 – y 4 if x = –3 and y = –2?
F. –7 G. –2 H. 2 I. 7
SOLUTION:
Replace x with –3 and y with –2 in the expression
x2 – y 4. Then write the powers as products and simplify.
The correct answer is F.
31. The table below shows the number of ants in an ant farm on different days. The number of ants doubles every ten days.
a. How many ants were in the farm on Day 1? b. How many ants will be in the farm on Day 91?
SOLUTION: a. Use the look for a pattern strategy. The number of ants doubles every 10 days, so going backward, the number of ants is halved each time the number of days is decreased by 10. Extend the pattern to find the number of ants on Day 1.
So, there were 10 ants in the farm on Day 1. b. Extend the pattern to find the number of ants in the farm on Day 91. The number of ants doubles every 10 days.
So, there will be 5,120 ants in the farm on Day 91.
Day 61 51 41 31 21 11 1 Number of ants
640 320 160 80 40 20 10
Day 51 61 71 81 91 Number of ants
320 640 1,280 2,560 5,120
32. Nieves and her three friends are playing a video game. The table shows their scores at the end of the first round.
a. What is the difference between the highest and lowest scores? b. By how many points is Nieves losing to Polly?
SOLUTION: a. The highest score is 230 points. The lowest score is –189. Subtract: 230 – (–189) = 230 + 189 or 419 The difference between the highest and lowest scores is 419 points. b. Nieves' score is –189. Polly's score is –142. Subtract: –142 – (–189) = –142 + 189 or 47 Polly is ahead of Nieves by 47 points.
Add. 33. –12 + (–19)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –12 + (–19) = –31
34. –8 + (–11)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –8 + (–11) = –19
35. –5 + 6
SOLUTION: Since the signs are different, subtract the absolute values of the numbers. Since the largest absolute value is 6, the sign of the answer is the same as the sign of the 6. –5 + 6 = 1
Write the expression using exponents. 1. (–5)(–5)(–5)(–5)
SOLUTION: The base –5 is a factor 4 times, so the exponent is 4.
(–5)(–5)(–5)(–5) = (–5) 4
2. 3 • 3 • 5 • q • q • q
SOLUTION: The base 3 is a factor 2 times, so its exponent is 2. The base 5 is a factor 1 time, so its exponent is 1. Because a number without an exponent is the same as a number with an exponent of 1, write 5 without an exponent. The base q is a factor 3 times, so its exponent is 3.
3 • 3 • 5 • q • q • q = 3 2 • 5 • q3
3. m • m • m • m • m
SOLUTION: The base m is a factor 5 times, so its exponent is 5.
m • m • m • m • m = m 5
Evaluate the expression.
4. (–9) 4
(–9) 4 = (–9) • (–9) • (–9) • (–9) or 6,561
5.
or
6.
7. In the United States, nearly 8 • 10 9 text messages
are sent every month. About how many text messages is this?
SOLUTION: Write the power as a product and multiply.
There are about 8 billion text messages sent in the United States each month.
8. Interstate 70 stretches almost 2 3 • 5
2 • 11 miles
across the United States. About how many miles long is Interstate 70?
SOLUTION: Write the power as a product and multiply.
Interstate 70 stretches for almost 2,200 miles.
Evaluate the expression.
9. g5 – h 3 if g = 2 and h = 7
SOLUTION: Replace g with 2 and h with 7 in the expression
. Then write the powers as products and simplify.
10. c 2 + d
3 if c = 8 and d = –3
SOLUTION: Replace c with 8 and d with –3 in the expression
. Then write the powers as products and simplify.
11. a 2 • b
SOLUTION:
Replace a with and b with 2 in the expression
. Then write the powers as products and simplify.
12. (r – s) 3 + r
2 if r = –3 and s = –4
SOLUTION: Replace r with –3 and s with –4 in the expression
. Then write the powers as products and simplify.
13. Model with Mathematics Refer to the graphic novel frame below for Exercises a–c.
The metric system is based on powers of 10. For example, one kilometer is equal to 1,000 meters or
10 3 meters. Write each measurement in meters as a
power of 10. a. megameter (1,000,000 meters) b. gigameter (1,000,000,000 meters) c. petameter (1,000,000,000,000,000 meters)
SOLUTION: a. megameter: 1,000,000 meters has 6 zeros or
powers of 10. So, a megameter is 10 6 .
b. gigameter: 1,000,000,000 meters has 9 zeros or
powers of 10. So, a gigameter is 10 9 .
c. petameter: 1,000,000,000,000,000 meters has 15
.
14. Identify Structure Write an expression with an exponent that has a value between 0 and 1.
SOLUTION:
3 = 27, 3
2 = 9, 3
1 = 3.
.
1 = 3
To go from 81 to 27, divide 81 by 3. This is the same for the rest of the values: 27 ÷ 3 = 9, and 9 ÷ 3 = 3.
To find the value of 2 -1
, follow the same pattern.
2 3 = 8, 2
0 = 2 ÷ 2 or
2 3 • 3
4
A. 3 • 3 • 4 • 4 • 4 B. 2 • 2 • 2 • 3 • 3 • 3 C. 2 • 2 • 2 • 3 • 3 • 3 • 3 D. 6 • 12
SOLUTION:
2 3 • 34
= 2 • 2 • 2 • 3 • 3 • 3 • 3. This corresponds to choice C.
17. Write 3 · p p p · 3 · 3 using exponents.
18. Evaluate x3 + y 4 if x = −3 and y = 4.
SOLUTION: Replace x with –3 and y with 4 in the expression
. Then write the powers as products and simplify.
Write the expression using exponents.
19.
SOLUTION:
The base is a factor 3 times, so its exponent is
3.
20. s • (–7) • s • (–7) • (–7)
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base –7 is a factor 3 times, so its exponent is 3. The base s is a factor 2 times, so its exponent is 2.
21. 4 • b • b • 4 • b • b
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base 4 is a factor 2 times, so its exponent is 2. The base b is a factor 4 times, so its exponent is 4.
Evaluate the expression.
SOLUTION:
Replace k with 3 and m with in the expression
k 4 • m. Then write the powers as products and
simplify.
3 , if c = –1 and d = 2
SOLUTION: Replace c with –1 and d with 2 in the expression
. Then write the powers as
products and simplify.
statement.
2
SOLUTION: Simplify each expression.
28 is larger than 25, so > will make the statement true.
(6 – 2) 2 + 3 • 4 > 5
2
3 3
SOLUTION: Simplify each expression.
81 is equal to 81, so = will make the statement true.
5 + 7 2 + 3
true.
=
27. Multiple Representations A square has a side length of s inches. a. Tables Copy and complete the table showing the side length, perimeter, and area of the square on a separate piece of paper.
b. Graphs On a separate piece of grid paper, graph the ordered pairs (side length, perimeter) and (side length, area) on the same coordinate plane. Then connect the points for each set. c. Words On a separate piece of paper, compare and contrast the graphs of the perimeter and area of the square. Which graph is a line?
SOLUTION: a. The perimeter of a square is found using the expression 4s. The area of a square is found using
the expression s 2 . Complete the table.
b. Graph the values from the table on the same coordinate plane. Side length is represented on the x- axis and both perimeter and area are represented on the y-axis.
c. Sample answer: The graph representing perimeter of a square is a line because each side length is multiplied by 4. The graph representing area of a square is nonlinear because each side length is squared. Squared values do not increase at a constant rate.
28. To find the volume of a cube, multiply its base, its height, and its width.
What is the volume of the cube expressed as a power?
A. 6 2 in
3
SOLUTION: A cube has the same measures for all sides, so the base, height, and width are all 6 inches. The volume
is 6 • 6 • 6 or 6 3 inches cubed. This corresponds to
choice B.
29. Short Response The volume of an ice cube in
cubic millimeters is represented by the term 11 3 .
What is 11 3 in standard form?
SOLUTION:
30. What is the value of x2 – y 4 if x = –3 and y = –2?
F. –7 G. –2 H. 2 I. 7
SOLUTION:
Replace x with –3 and y with –2 in the expression
x2 – y 4. Then write the powers as products and simplify.
The correct answer is F.
31. The table below shows the number of ants in an ant farm on different days. The number of ants doubles every ten days.
a. How many ants were in the farm on Day 1? b. How many ants will be in the farm on Day 91?
SOLUTION: a. Use the look for a pattern strategy. The number of ants doubles every 10 days, so going backward, the number of ants is halved each time the number of days is decreased by 10. Extend the pattern to find the number of ants on Day 1.
So, there were 10 ants in the farm on Day 1. b. Extend the pattern to find the number of ants in the farm on Day 91. The number of ants doubles every 10 days.
So, there will be 5,120 ants in the farm on Day 91.
Day 61 51 41 31 21 11 1 Number of ants
640 320 160 80 40 20 10
Day 51 61 71 81 91 Number of ants
320 640 1,280 2,560 5,120
32. Nieves and her three friends are playing a video game. The table shows their scores at the end of the first round.
a. What is the difference between the highest and lowest scores? b. By how many points is Nieves losing to Polly?
SOLUTION: a. The highest score is 230 points. The lowest score is –189. Subtract: 230 – (–189) = 230 + 189 or 419 The difference between the highest and lowest scores is 419 points. b. Nieves' score is –189. Polly's score is –142. Subtract: –142 – (–189) = –142 + 189 or 47 Polly is ahead of Nieves by 47 points.
Add. 33. –12 + (–19)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –12 + (–19) = –31
34. –8 + (–11)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –8 + (–11) = –19
35. –5 + 6
SOLUTION: Since the signs are different, subtract the absolute values of the numbers. Since the largest absolute value is 6, the sign of the answer is the same as the sign of the 6. –5 + 6 = 1
eSolutions Manual - Powered by Cognero Page 1
1-2 Powers and Exponents
Write the expression using exponents. 1. (–5)(–5)(–5)(–5)
SOLUTION: The base –5 is a factor 4 times, so the exponent is 4.
(–5)(–5)(–5)(–5) = (–5) 4
2. 3 • 3 • 5 • q • q • q
SOLUTION: The base 3 is a factor 2 times, so its exponent is 2. The base 5 is a factor 1 time, so its exponent is 1. Because a number without an exponent is the same as a number with an exponent of 1, write 5 without an exponent. The base q is a factor 3 times, so its exponent is 3.
3 • 3 • 5 • q • q • q = 3 2 • 5 • q3
3. m • m • m • m • m
SOLUTION: The base m is a factor 5 times, so its exponent is 5.
m • m • m • m • m = m 5
Evaluate the expression.
4. (–9) 4
(–9) 4 = (–9) • (–9) • (–9) • (–9) or 6,561
5.
or
6.
7. In the United States, nearly 8 • 10 9 text messages
are sent every month. About how many text messages is this?
SOLUTION: Write the power as a product and multiply.
There are about 8 billion text messages sent in the United States each month.
8. Interstate 70 stretches almost 2 3 • 5
2 • 11 miles
across the United States. About how many miles long is Interstate 70?
SOLUTION: Write the power as a product and multiply.
Interstate 70 stretches for almost 2,200 miles.
Evaluate the expression.
9. g5 – h 3 if g = 2 and h = 7
SOLUTION: Replace g with 2 and h with 7 in the expression
. Then write the powers as products and simplify.
10. c 2 + d
3 if c = 8 and d = –3
SOLUTION: Replace c with 8 and d with –3 in the expression
. Then write the powers as products and simplify.
11. a 2 • b
SOLUTION:
Replace a with and b with 2 in the expression
. Then write the powers as products and simplify.
12. (r – s) 3 + r
2 if r = –3 and s = –4
SOLUTION: Replace r with –3 and s with –4 in the expression
. Then write the powers as products and simplify.
13. Model with Mathematics Refer to the graphic novel frame below for Exercises a–c.
The metric system is based on powers of 10. For example, one kilometer is equal to 1,000 meters or
10 3 meters. Write each measurement in meters as a
power of 10. a. megameter (1,000,000 meters) b. gigameter (1,000,000,000 meters) c. petameter (1,000,000,000,000,000 meters)
SOLUTION: a. megameter: 1,000,000 meters has 6 zeros or
powers of 10. So, a megameter is 10 6 .
b. gigameter: 1,000,000,000 meters has 9 zeros or
powers of 10. So, a gigameter is 10 9 .
c. petameter: 1,000,000,000,000,000 meters has 15
.
14. Identify Structure Write an expression with an exponent that has a value between 0 and 1.
SOLUTION:
3 = 27, 3
2 = 9, 3
1 = 3.
.
1 = 3
To go from 81 to 27, divide 81 by 3. This is the same for the rest of the values: 27 ÷ 3 = 9, and 9 ÷ 3 = 3.
To find the value of 2 -1
, follow the same pattern.
2 3 = 8, 2
0 = 2 ÷ 2 or
2 3 • 3
4
A. 3 • 3 • 4 • 4 • 4 B. 2 • 2 • 2 • 3 • 3 • 3 C. 2 • 2 • 2 • 3 • 3 • 3 • 3 D. 6 • 12
SOLUTION:
2 3 • 34
= 2 • 2 • 2 • 3 • 3 • 3 • 3. This corresponds to choice C.
17. Write 3 · p p p · 3 · 3 using exponents.
18. Evaluate x3 + y 4 if x = −3 and y = 4.
SOLUTION: Replace x with –3 and y with 4 in the expression
. Then write the powers as products and simplify.
Write the expression using exponents.
19.
SOLUTION:
The base is a factor 3 times, so its exponent is
3.
20. s • (–7) • s • (–7) • (–7)
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base –7 is a factor 3 times, so its exponent is 3. The base s is a factor 2 times, so its exponent is 2.
21. 4 • b • b • 4 • b • b
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base 4 is a factor 2 times, so its exponent is 2. The base b is a factor 4 times, so its exponent is 4.
Evaluate the expression.
SOLUTION:
Replace k with 3 and m with in the expression
k 4 • m. Then write the powers as products and
simplify.
3 , if c = –1 and d = 2
SOLUTION: Replace c with –1 and d with 2 in the expression
. Then write the powers as
products and simplify.
statement.
2
SOLUTION: Simplify each expression.
28 is larger than 25, so > will make the statement true.
(6 – 2) 2 + 3 • 4 > 5
2
3 3
SOLUTION: Simplify each expression.
81 is equal to 81, so = will make the statement true.
5 + 7 2 + 3
true.
=
27. Multiple Representations A square has a side length of s inches. a. Tables Copy and complete the table showing the side length, perimeter, and area of the square on a separate piece of paper.
b. Graphs On a separate piece of grid paper, graph the ordered pairs (side length, perimeter) and (side length, area) on the same coordinate plane. Then connect the points for each set. c. Words On a separate piece of paper, compare and contrast the graphs of the perimeter and area of the square. Which graph is a line?
SOLUTION: a. The perimeter of a square is found using the expression 4s. The area of a square is found using
the expression s 2 . Complete the table.
b. Graph the values from the table on the same coordinate plane. Side length is represented on the x- axis and both perimeter and area are represented on the y-axis.
c. Sample answer: The graph representing perimeter of a square is a line because each side length is multiplied by 4. The graph representing area of a square is nonlinear because each side length is squared. Squared values do not increase at a constant rate.
28. To find the volume of a cube, multiply its base, its height, and its width.
What is the volume of the cube expressed as a power?
A. 6 2 in
3
SOLUTION: A cube has the same measures for all sides, so the base, height, and width are all 6 inches. The volume
is 6 • 6 • 6 or 6 3 inches cubed. This corresponds to
choice B.
29. Short Response The volume of an ice cube in
cubic millimeters is represented by the term 11 3 .
What is 11 3 in standard form?
SOLUTION:
30. What is the value of x2 – y 4 if x = –3 and y = –2?
F. –7 G. –2 H. 2 I. 7
SOLUTION:
Replace x with –3 and y with –2 in the expression
x2 – y 4. Then write the powers as products and simplify.
The correct answer is F.
31. The table below shows the number of ants in an ant farm on different days. The number of ants doubles every ten days.
a. How many ants were in the farm on Day 1? b. How many ants will be in the farm on Day 91?
SOLUTION: a. Use the look for a pattern strategy. The number of ants doubles every 10 days, so going backward, the number of ants is halved each time the number of days is decreased by 10. Extend the pattern to find the number of ants on Day 1.
So, there were 10 ants in the farm on Day 1. b. Extend the pattern to find the number of ants in the farm on Day 91. The number of ants doubles every 10 days.
So, there will be 5,120 ants in the farm on Day 91.
Day 61 51 41 31 21 11 1 Number of ants
640 320 160 80 40 20 10
Day 51 61 71 81 91 Number of ants
320 640 1,280 2,560 5,120
32. Nieves and her three friends are playing a video game. The table shows their scores at the end of the first round.
a. What is the difference between the highest and lowest scores? b. By how many points is Nieves losing to Polly?
SOLUTION: a. The highest score is 230 points. The lowest score is –189. Subtract: 230 – (–189) = 230 + 189 or 419 The difference between the highest and lowest scores is 419 points. b. Nieves' score is –189. Polly's score is –142. Subtract: –142 – (–189) = –142 + 189 or 47 Polly is ahead of Nieves by 47 points.
Add. 33. –12 + (–19)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –12 + (–19) = –31
34. –8 + (–11)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –8 + (–11) = –19
35. –5 + 6
SOLUTION: Since the signs are different, subtract the absolute values of the numbers. Since the largest absolute value is 6, the sign of the answer is the same as the sign of the 6. –5 + 6 = 1
Write the expression using exponents. 1. (–5)(–5)(–5)(–5)
SOLUTION: The base –5 is a factor 4 times, so the exponent is 4.
(–5)(–5)(–5)(–5) = (–5) 4
2. 3 • 3 • 5 • q • q • q
SOLUTION: The base 3 is a factor 2 times, so its exponent is 2. The base 5 is a factor 1 time, so its exponent is 1. Because a number without an exponent is the same as a number with an exponent of 1, write 5 without an exponent. The base q is a factor 3 times, so its exponent is 3.
3 • 3 • 5 • q • q • q = 3 2 • 5 • q3
3. m • m • m • m • m
SOLUTION: The base m is a factor 5 times, so its exponent is 5.
m • m • m • m • m = m 5
Evaluate the expression.
4. (–9) 4
(–9) 4 = (–9) • (–9) • (–9) • (–9) or 6,561
5.
or
6.
7. In the United States, nearly 8 • 10 9 text messages
are sent every month. About how many text messages is this?
SOLUTION: Write the power as a product and multiply.
There are about 8 billion text messages sent in the United States each month.
8. Interstate 70 stretches almost 2 3 • 5
2 • 11 miles
across the United States. About how many miles long is Interstate 70?
SOLUTION: Write the power as a product and multiply.
Interstate 70 stretches for almost 2,200 miles.
Evaluate the expression.
9. g5 – h 3 if g = 2 and h = 7
SOLUTION: Replace g with 2 and h with 7 in the expression
. Then write the powers as products and simplify.
10. c 2 + d
3 if c = 8 and d = –3
SOLUTION: Replace c with 8 and d with –3 in the expression
. Then write the powers as products and simplify.
11. a 2 • b
SOLUTION:
Replace a with and b with 2 in the expression
. Then write the powers as products and simplify.
12. (r – s) 3 + r
2 if r = –3 and s = –4
SOLUTION: Replace r with –3 and s with –4 in the expression
. Then write the powers as products and simplify.
13. Model with Mathematics Refer to the graphic novel frame below for Exercises a–c.
The metric system is based on powers of 10. For example, one kilometer is equal to 1,000 meters or
10 3 meters. Write each measurement in meters as a
power of 10. a. megameter (1,000,000 meters) b. gigameter (1,000,000,000 meters) c. petameter (1,000,000,000,000,000 meters)
SOLUTION: a. megameter: 1,000,000 meters has 6 zeros or
powers of 10. So, a megameter is 10 6 .
b. gigameter: 1,000,000,000 meters has 9 zeros or
powers of 10. So, a gigameter is 10 9 .
c. petameter: 1,000,000,000,000,000 meters has 15
.
14. Identify Structure Write an expression with an exponent that has a value between 0 and 1.
SOLUTION:
3 = 27, 3
2 = 9, 3
1 = 3.
.
1 = 3
To go from 81 to 27, divide 81 by 3. This is the same for the rest of the values: 27 ÷ 3 = 9, and 9 ÷ 3 = 3.
To find the value of 2 -1
, follow the same pattern.
2 3 = 8, 2
0 = 2 ÷ 2 or
2 3 • 3
4
A. 3 • 3 • 4 • 4 • 4 B. 2 • 2 • 2 • 3 • 3 • 3 C. 2 • 2 • 2 • 3 • 3 • 3 • 3 D. 6 • 12
SOLUTION:
2 3 • 34
= 2 • 2 • 2 • 3 • 3 • 3 • 3. This corresponds to choice C.
17. Write 3 · p p p · 3 · 3 using exponents.
18. Evaluate x3 + y 4 if x = −3 and y = 4.
SOLUTION: Replace x with –3 and y with 4 in the expression
. Then write the powers as products and simplify.
Write the expression using exponents.
19.
SOLUTION:
The base is a factor 3 times, so its exponent is
3.
20. s • (–7) • s • (–7) • (–7)
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base –7 is a factor 3 times, so its exponent is 3. The base s is a factor 2 times, so its exponent is 2.
21. 4 • b • b • 4 • b • b
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base 4 is a factor 2 times, so its exponent is 2. The base b is a factor 4 times, so its exponent is 4.
Evaluate the expression.
SOLUTION:
Replace k with 3 and m with in the expression
k 4 • m. Then write the powers as products and
simplify.
3 , if c = –1 and d = 2
SOLUTION: Replace c with –1 and d with 2 in the expression
. Then write the powers as
products and simplify.
statement.
2
SOLUTION: Simplify each expression.
28 is larger than 25, so > will make the statement true.
(6 – 2) 2 + 3 • 4 > 5
2
3 3
SOLUTION: Simplify each expression.
81 is equal to 81, so = will make the statement true.
5 + 7 2 + 3
true.
=
27. Multiple Representations A square has a side length of s inches. a. Tables Copy and complete the table showing the side length, perimeter, and area of the square on a separate piece of paper.
b. Graphs On a separate piece of grid paper, graph the ordered pairs (side length, perimeter) and (side length, area) on the same coordinate plane. Then connect the points for each set. c. Words On a separate piece of paper, compare and contrast the graphs of the perimeter and area of the square. Which graph is a line?
SOLUTION: a. The perimeter of a square is found using the expression 4s. The area of a square is found using
the expression s 2 . Complete the table.
b. Graph the values from the table on the same coordinate plane. Side length is represented on the x- axis and both perimeter and area are represented on the y-axis.
c. Sample answer: The graph representing perimeter of a square is a line because each side length is multiplied by 4. The graph representing area of a square is nonlinear because each side length is squared. Squared values do not increase at a constant rate.
28. To find the volume of a cube, multiply its base, its height, and its width.
What is the volume of the cube expressed as a power?
A. 6 2 in
3
SOLUTION: A cube has the same measures for all sides, so the base, height, and width are all 6 inches. The volume
is 6 • 6 • 6 or 6 3 inches cubed. This corresponds to
choice B.
29. Short Response The volume of an ice cube in
cubic millimeters is represented by the term 11 3 .
What is 11 3 in standard form?
SOLUTION:
30. What is the value of x2 – y 4 if x = –3 and y = –2?
F. –7 G. –2 H. 2 I. 7
SOLUTION:
Replace x with –3 and y with –2 in the expression
x2 – y 4. Then write the powers as products and simplify.
The correct answer is F.
31. The table below shows the number of ants in an ant farm on different days. The number of ants doubles every ten days.
a. How many ants were in the farm on Day 1? b. How many ants will be in the farm on Day 91?
SOLUTION: a. Use the look for a pattern strategy. The number of ants doubles every 10 days, so going backward, the number of ants is halved each time the number of days is decreased by 10. Extend the pattern to find the number of ants on Day 1.
So, there were 10 ants in the farm on Day 1. b. Extend the pattern to find the number of ants in the farm on Day 91. The number of ants doubles every 10 days.
So, there will be 5,120 ants in the farm on Day 91.
Day 61 51 41 31 21 11 1 Number of ants
640 320 160 80 40 20 10
Day 51 61 71 81 91 Number of ants
320 640 1,280 2,560 5,120
32. Nieves and her three friends are playing a video game. The table shows their scores at the end of the first round.
a. What is the difference between the highest and lowest scores? b. By how many points is Nieves losing to Polly?
SOLUTION: a. The highest score is 230 points. The lowest score is –189. Subtract: 230 – (–189) = 230 + 189 or 419 The difference between the highest and lowest scores is 419 points. b. Nieves' score is –189. Polly's score is –142. Subtract: –142 – (–189) = –142 + 189 or 47 Polly is ahead of Nieves by 47 points.
Add. 33. –12 + (–19)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –12 + (–19) = –31
34. –8 + (–11)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –8 + (–11) = –19
35. –5 + 6
SOLUTION: Since the signs are different, subtract the absolute values of the numbers. Since the largest absolute value is 6, the sign of the answer is the same as the sign of the 6. –5 + 6 = 1
eSolutions Manual - Powered by Cognero Page 2
1-2 Powers and Exponents
Write the expression using exponents. 1. (–5)(–5)(–5)(–5)
SOLUTION: The base –5 is a factor 4 times, so the exponent is 4.
(–5)(–5)(–5)(–5) = (–5) 4
2. 3 • 3 • 5 • q • q • q
SOLUTION: The base 3 is a factor 2 times, so its exponent is 2. The base 5 is a factor 1 time, so its exponent is 1. Because a number without an exponent is the same as a number with an exponent of 1, write 5 without an exponent. The base q is a factor 3 times, so its exponent is 3.
3 • 3 • 5 • q • q • q = 3 2 • 5 • q3
3. m • m • m • m • m
SOLUTION: The base m is a factor 5 times, so its exponent is 5.
m • m • m • m • m = m 5
Evaluate the expression.
4. (–9) 4
(–9) 4 = (–9) • (–9) • (–9) • (–9) or 6,561
5.
or
6.
7. In the United States, nearly 8 • 10 9 text messages
are sent every month. About how many text messages is this?
SOLUTION: Write the power as a product and multiply.
There are about 8 billion text messages sent in the United States each month.
8. Interstate 70 stretches almost 2 3 • 5
2 • 11 miles
across the United States. About how many miles long is Interstate 70?
SOLUTION: Write the power as a product and multiply.
Interstate 70 stretches for almost 2,200 miles.
Evaluate the expression.
9. g5 – h 3 if g = 2 and h = 7
SOLUTION: Replace g with 2 and h with 7 in the expression
. Then write the powers as products and simplify.
10. c 2 + d
3 if c = 8 and d = –3
SOLUTION: Replace c with 8 and d with –3 in the expression
. Then write the powers as products and simplify.
11. a 2 • b
SOLUTION:
Replace a with and b with 2 in the expression
. Then write the powers as products and simplify.
12. (r – s) 3 + r
2 if r = –3 and s = –4
SOLUTION: Replace r with –3 and s with –4 in the expression
. Then write the powers as products and simplify.
13. Model with Mathematics Refer to the graphic novel frame below for Exercises a–c.
The metric system is based on powers of 10. For example, one kilometer is equal to 1,000 meters or
10 3 meters. Write each measurement in meters as a
power of 10. a. megameter (1,000,000 meters) b. gigameter (1,000,000,000 meters) c. petameter (1,000,000,000,000,000 meters)
SOLUTION: a. megameter: 1,000,000 meters has 6 zeros or
powers of 10. So, a megameter is 10 6 .
b. gigameter: 1,000,000,000 meters has 9 zeros or
powers of 10. So, a gigameter is 10 9 .
c. petameter: 1,000,000,000,000,000 meters has 15
.
14. Identify Structure Write an expression with an exponent that has a value between 0 and 1.
SOLUTION:
3 = 27, 3
2 = 9, 3
1 = 3.
.
1 = 3
To go from 81 to 27, divide 81 by 3. This is the same for the rest of the values: 27 ÷ 3 = 9, and 9 ÷ 3 = 3.
To find the value of 2 -1
, follow the same pattern.
2 3 = 8, 2
0 = 2 ÷ 2 or
2 3 • 3
4
A. 3 • 3 • 4 • 4 • 4 B. 2 • 2 • 2 • 3 • 3 • 3 C. 2 • 2 • 2 • 3 • 3 • 3 • 3 D. 6 • 12
SOLUTION:
2 3 • 34
= 2 • 2 • 2 • 3 • 3 • 3 • 3. This corresponds to choice C.
17. Write 3 · p p p · 3 · 3 using exponents.
18. Evaluate x3 + y 4 if x = −3 and y = 4.
SOLUTION: Replace x with –3 and y with 4 in the expression
. Then write the powers as products and simplify.
Write the expression using exponents.
19.
SOLUTION:
The base is a factor 3 times, so its exponent is
3.
20. s • (–7) • s • (–7) • (–7)
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base –7 is a factor 3 times, so its exponent is 3. The base s is a factor 2 times, so its exponent is 2.
21. 4 • b • b • 4 • b • b
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base 4 is a factor 2 times, so its exponent is 2. The base b is a factor 4 times, so its exponent is 4.
Evaluate the expression.
SOLUTION:
Replace k with 3 and m with in the expression
k 4 • m. Then write the powers as products and
simplify.
3 , if c = –1 and d = 2
SOLUTION: Replace c with –1 and d with 2 in the expression
. Then write the powers as
products and simplify.
statement.
2
SOLUTION: Simplify each expression.
28 is larger than 25, so > will make the statement true.
(6 – 2) 2 + 3 • 4 > 5
2
3 3
SOLUTION: Simplify each expression.
81 is equal to 81, so = will make the statement true.
5 + 7 2 + 3
true.
=
27. Multiple Representations A square has a side length of s inches. a. Tables Copy and complete the table showing the side length, perimeter, and area of the square on a separate piece of paper.
b. Graphs On a separate piece of grid paper, graph the ordered pairs (side length, perimeter) and (side length, area) on the same coordinate plane. Then connect the points for each set. c. Words On a separate piece of paper, compare and contrast the graphs of the perimeter and area of the square. Which graph is a line?
SOLUTION: a. The perimeter of a square is found using the expression 4s. The area of a square is found using
the expression s 2 . Complete the table.
b. Graph the values from the table on the same coordinate plane. Side length is represented on the x- axis and both perimeter and area are represented on the y-axis.
c. Sample answer: The graph representing perimeter of a square is a line because each side length is multiplied by 4. The graph representing area of a square is nonlinear because each side length is squared. Squared values do not increase at a constant rate.
28. To find the volume of a cube, multiply its base, its height, and its width.
What is the volume of the cube expressed as a power?
A. 6 2 in
3
SOLUTION: A cube has the same measures for all sides, so the base, height, and width are all 6 inches. The volume
is 6 • 6 • 6 or 6 3 inches cubed. This corresponds to
choice B.
29. Short Response The volume of an ice cube in
cubic millimeters is represented by the term 11 3 .
What is 11 3 in standard form?
SOLUTION:
30. What is the value of x2 – y 4 if x = –3 and y = –2?
F. –7 G. –2 H. 2 I. 7
SOLUTION:
Replace x with –3 and y with –2 in the expression
x2 – y 4. Then write the powers as products and simplify.
The correct answer is F.
31. The table below shows the number of ants in an ant farm on different days. The number of ants doubles every ten days.
a. How many ants were in the farm on Day 1? b. How many ants will be in the farm on Day 91?
SOLUTION: a. Use the look for a pattern strategy. The number of ants doubles every 10 days, so going backward, the number of ants is halved each time the number of days is decreased by 10. Extend the pattern to find the number of ants on Day 1.
So, there were 10 ants in the farm on Day 1. b. Extend the pattern to find the number of ants in the farm on Day 91. The number of ants doubles every 10 days.
So, there will be 5,120 ants in the farm on Day 91.
Day 61 51 41 31 21 11 1 Number of ants
640 320 160 80 40 20 10
Day 51 61 71 81 91 Number of ants
320 640 1,280 2,560 5,120
32. Nieves and her three friends are playing a video game. The table shows their scores at the end of the first round.
a. What is the difference between the highest and lowest scores? b. By how many points is Nieves losing to Polly?
SOLUTION: a. The highest score is 230 points. The lowest score is –189. Subtract: 230 – (–189) = 230 + 189 or 419 The difference between the highest and lowest scores is 419 points. b. Nieves' score is –189. Polly's score is –142. Subtract: –142 – (–189) = –142 + 189 or 47 Polly is ahead of Nieves by 47 points.
Add. 33. –12 + (–19)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –12 + (–19) = –31
34. –8 + (–11)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –8 + (–11) = –19
35. –5 + 6
SOLUTION: Since the signs are different, subtract the absolute values of the numbers. Since the largest absolute value is 6, the sign of the answer is the same as the sign of the 6. –5 + 6 = 1
Write the expression using exponents. 1. (–5)(–5)(–5)(–5)
SOLUTION: The base –5 is a factor 4 times, so the exponent is 4.
(–5)(–5)(–5)(–5) = (–5) 4
2. 3 • 3 • 5 • q • q • q
SOLUTION: The base 3 is a factor 2 times, so its exponent is 2. The base 5 is a factor 1 time, so its exponent is 1. Because a number without an exponent is the same as a number with an exponent of 1, write 5 without an exponent. The base q is a factor 3 times, so its exponent is 3.
3 • 3 • 5 • q • q • q = 3 2 • 5 • q3
3. m • m • m • m • m
SOLUTION: The base m is a factor 5 times, so its exponent is 5.
m • m • m • m • m = m 5
Evaluate the expression.
4. (–9) 4
(–9) 4 = (–9) • (–9) • (–9) • (–9) or 6,561
5.
or
6.
7. In the United States, nearly 8 • 10 9 text messages
are sent every month. About how many text messages is this?
SOLUTION: Write the power as a product and multiply.
There are about 8 billion text messages sent in the United States each month.
8. Interstate 70 stretches almost 2 3 • 5
2 • 11 miles
across the United States. About how many miles long is Interstate 70?
SOLUTION: Write the power as a product and multiply.
Interstate 70 stretches for almost 2,200 miles.
Evaluate the expression.
9. g5 – h 3 if g = 2 and h = 7
SOLUTION: Replace g with 2 and h with 7 in the expression
. Then write the powers as products and simplify.
10. c 2 + d
3 if c = 8 and d = –3
SOLUTION: Replace c with 8 and d with –3 in the expression
. Then write the powers as products and simplify.
11. a 2 • b
SOLUTION:
Replace a with and b with 2 in the expression
. Then write the powers as products and simplify.
12. (r – s) 3 + r
2 if r = –3 and s = –4
SOLUTION: Replace r with –3 and s with –4 in the expression
. Then write the powers as products and simplify.
13. Model with Mathematics Refer to the graphic novel frame below for Exercises a–c.
The metric system is based on powers of 10. For example, one kilometer is equal to 1,000 meters or
10 3 meters. Write each measurement in meters as a
power of 10. a. megameter (1,000,000 meters) b. gigameter (1,000,000,000 meters) c. petameter (1,000,000,000,000,000 meters)
SOLUTION: a. megameter: 1,000,000 meters has 6 zeros or
powers of 10. So, a megameter is 10 6 .
b. gigameter: 1,000,000,000 meters has 9 zeros or
powers of 10. So, a gigameter is 10 9 .
c. petameter: 1,000,000,000,000,000 meters has 15
.
14. Identify Structure Write an expression with an exponent that has a value between 0 and 1.
SOLUTION:
3 = 27, 3
2 = 9, 3
1 = 3.
.
1 = 3
To go from 81 to 27, divide 81 by 3. This is the same for the rest of the values: 27 ÷ 3 = 9, and 9 ÷ 3 = 3.
To find the value of 2 -1
, follow the same pattern.
2 3 = 8, 2
0 = 2 ÷ 2 or
2 3 • 3
4
A. 3 • 3 • 4 • 4 • 4 B. 2 • 2 • 2 • 3 • 3 • 3 C. 2 • 2 • 2 • 3 • 3 • 3 • 3 D. 6 • 12
SOLUTION:
2 3 • 34
= 2 • 2 • 2 • 3 • 3 • 3 • 3. This corresponds to choice C.
17. Write 3 · p p p · 3 · 3 using exponents.
18. Evaluate x3 + y 4 if x = −3 and y = 4.
SOLUTION: Replace x with –3 and y with 4 in the expression
. Then write the powers as products and simplify.
Write the expression using exponents.
19.
SOLUTION:
The base is a factor 3 times, so its exponent is
3.
20. s • (–7) • s • (–7) • (–7)
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base –7 is a factor 3 times, so its exponent is 3. The base s is a factor 2 times, so its exponent is 2.
21. 4 • b • b • 4 • b • b
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base 4 is a factor 2 times, so its exponent is 2. The base b is a factor 4 times, so its exponent is 4.
Evaluate the expression.
SOLUTION:
Replace k with 3 and m with in the expression
k 4 • m. Then write the powers as products and
simplify.
3 , if c = –1 and d = 2
SOLUTION: Replace c with –1 and d with 2 in the expression
. Then write the powers as
products and simplify.
statement.
2
SOLUTION: Simplify each expression.
28 is larger than 25, so > will make the statement true.
(6 – 2) 2 + 3 • 4 > 5
2
3 3
SOLUTION: Simplify each expression.
81 is equal to 81, so = will make the statement true.
5 + 7 2 + 3
true.
=
27. Multiple Representations A square has a side length of s inches. a. Tables Copy and complete the table showing the side length, perimeter, and area of the square on a separate piece of paper.
b. Graphs On a separate piece of grid paper, graph the ordered pairs (side length, perimeter) and (side length, area) on the same coordinate plane. Then connect the points for each set. c. Words On a separate piece of paper, compare and contrast the graphs of the perimeter and area of the square. Which graph is a line?
SOLUTION: a. The perimeter of a square is found using the expression 4s. The area of a square is found using
the expression s 2 . Complete the table.
b. Graph the values from the table on the same coordinate plane. Side length is represented on the x- axis and both perimeter and area are represented on the y-axis.
c. Sample answer: The graph representing perimeter of a square is a line because each side length is multiplied by 4. The graph representing area of a square is nonlinear because each side length is squared. Squared values do not increase at a constant rate.
28. To find the volume of a cube, multiply its base, its height, and its width.
What is the volume of the cube expressed as a power?
A. 6 2 in
3
SOLUTION: A cube has the same measures for all sides, so the base, height, and width are all 6 inches. The volume
is 6 • 6 • 6 or 6 3 inches cubed. This corresponds to
choice B.
29. Short Response The volume of an ice cube in
cubic millimeters is represented by the term 11 3 .
What is 11 3 in standard form?
SOLUTION:
30. What is the value of x2 – y 4 if x = –3 and y = –2?
F. –7 G. –2 H. 2 I. 7
SOLUTION:
Replace x with –3 and y with –2 in the expression
x2 – y 4. Then write the powers as products and simplify.
The correct answer is F.
31. The table below shows the number of ants in an ant farm on different days. The number of ants doubles every ten days.
a. How many ants were in the farm on Day 1? b. How many ants will be in the farm on Day 91?
SOLUTION: a. Use the look for a pattern strategy. The number of ants doubles every 10 days, so going backward, the number of ants is halved each time the number of days is decreased by 10. Extend the pattern to find the number of ants on Day 1.
So, there were 10 ants in the farm on Day 1. b. Extend the pattern to find the number of ants in the farm on Day 91. The number of ants doubles every 10 days.
So, there will be 5,120 ants in the farm on Day 91.
Day 61 51 41 31 21 11 1 Number of ants
640 320 160 80 40 20 10
Day 51 61 71 81 91 Number of ants
320 640 1,280 2,560 5,120
32. Nieves and her three friends are playing a video game. The table shows their scores at the end of the first round.
a. What is the difference between the highest and lowest scores? b. By how many points is Nieves losing to Polly?
SOLUTION: a. The highest score is 230 points. The lowest score is –189. Subtract: 230 – (–189) = 230 + 189 or 419 The difference between the highest and lowest scores is 419 points. b. Nieves' score is –189. Polly's score is –142. Subtract: –142 – (–189) = –142 + 189 or 47 Polly is ahead of Nieves by 47 points.
Add. 33. –12 + (–19)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –12 + (–19) = –31
34. –8 + (–11)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –8 + (–11) = –19
35. –5 + 6
SOLUTION: Since the signs are different, subtract the absolute values of the numbers. Since the largest absolute value is 6, the sign of the answer is the same as the sign of the 6. –5 + 6 = 1
eSolutions Manual - Powered by Cognero Page 3
1-2 Powers and Exponents
Write the expression using exponents. 1. (–5)(–5)(–5)(–5)
SOLUTION: The base –5 is a factor 4 times, so the exponent is 4.
(–5)(–5)(–5)(–5) = (–5) 4
2. 3 • 3 • 5 • q • q • q
SOLUTION: The base 3 is a factor 2 times, so its exponent is 2. The base 5 is a factor 1 time, so its exponent is 1. Because a number without an exponent is the same as a number with an exponent of 1, write 5 without an exponent. The base q is a factor 3 times, so its exponent is 3.
3 • 3 • 5 • q • q • q = 3 2 • 5 • q3
3. m • m • m • m • m
SOLUTION: The base m is a factor 5 times, so its exponent is 5.
m • m • m • m • m = m 5
Evaluate the expression.
4. (–9) 4
(–9) 4 = (–9) • (–9) • (–9) • (–9) or 6,561
5.
or
6.
7. In the United States, nearly 8 • 10 9 text messages
are sent every month. About how many text messages is this?
SOLUTION: Write the power as a product and multiply.
There are about 8 billion text messages sent in the United States each month.
8. Interstate 70 stretches almost 2 3 • 5
2 • 11 miles
across the United States. About how many miles long is Interstate 70?
SOLUTION: Write the power as a product and multiply.
Interstate 70 stretches for almost 2,200 miles.
Evaluate the expression.
9. g5 – h 3 if g = 2 and h = 7
SOLUTION: Replace g with 2 and h with 7 in the expression
. Then write the powers as products and simplify.
10. c 2 + d
3 if c = 8 and d = –3
SOLUTION: Replace c with 8 and d with –3 in the expression
. Then write the powers as products and simplify.
11. a 2 • b
SOLUTION:
Replace a with and b with 2 in the expression
. Then write the powers as products and simplify.
12. (r – s) 3 + r
2 if r = –3 and s = –4
SOLUTION: Replace r with –3 and s with –4 in the expression
. Then write the powers as products and simplify.
13. Model with Mathematics Refer to the graphic novel frame below for Exercises a–c.
The metric system is based on powers of 10. For example, one kilometer is equal to 1,000 meters or
10 3 meters. Write each measurement in meters as a
power of 10. a. megameter (1,000,000 meters) b. gigameter (1,000,000,000 meters) c. petameter (1,000,000,000,000,000 meters)
SOLUTION: a. megameter: 1,000,000 meters has 6 zeros or
powers of 10. So, a megameter is 10 6 .
b. gigameter: 1,000,000,000 meters has 9 zeros or
powers of 10. So, a gigameter is 10 9 .
c. petameter: 1,000,000,000,000,000 meters has 15
.
14. Identify Structure Write an expression with an exponent that has a value between 0 and 1.
SOLUTION:
3 = 27, 3
2 = 9, 3
1 = 3.
.
1 = 3
To go from 81 to 27, divide 81 by 3. This is the same for the rest of the values: 27 ÷ 3 = 9, and 9 ÷ 3 = 3.
To find the value of 2 -1
, follow the same pattern.
2 3 = 8, 2
0 = 2 ÷ 2 or
2 3 • 3
4
A. 3 • 3 • 4 • 4 • 4 B. 2 • 2 • 2 • 3 • 3 • 3 C. 2 • 2 • 2 • 3 • 3 • 3 • 3 D. 6 • 12
SOLUTION:
2 3 • 34
= 2 • 2 • 2 • 3 • 3 • 3 • 3. This corresponds to choice C.
17. Write 3 · p p p · 3 · 3 using exponents.
18. Evaluate x3 + y 4 if x = −3 and y = 4.
SOLUTION: Replace x with –3 and y with 4 in the expression
. Then write the powers as products and simplify.
Write the expression using exponents.
19.
SOLUTION:
The base is a factor 3 times, so its exponent is
3.
20. s • (–7) • s • (–7) • (–7)
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base –7 is a factor 3 times, so its exponent is 3. The base s is a factor 2 times, so its exponent is 2.
21. 4 • b • b • 4 • b • b
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base 4 is a factor 2 times, so its exponent is 2. The base b is a factor 4 times, so its exponent is 4.
Evaluate the expression.
SOLUTION:
Replace k with 3 and m with in the expression
k 4 • m. Then write the powers as products and
simplify.
3 , if c = –1 and d = 2
SOLUTION: Replace c with –1 and d with 2 in the expression
. Then write the powers as
products and simplify.
statement.
2
SOLUTION: Simplify each expression.
28 is larger than 25, so > will make the statement true.
(6 – 2) 2 + 3 • 4 > 5
2
3 3
SOLUTION: Simplify each expression.
81 is equal to 81, so = will make the statement true.
5 + 7 2 + 3
true.
=
27. Multiple Representations A square has a side length of s inches. a. Tables Copy and complete the table showing the side length, perimeter, and area of the square on a separate piece of paper.
b. Graphs On a separate piece of grid paper, graph the ordered pairs (side length, perimeter) and (side length, area) on the same coordinate plane. Then connect the points for each set. c. Words On a separate piece of paper, compare and contrast the graphs of the perimeter and area of the square. Which graph is a line?
SOLUTION: a. The perimeter of a square is found using the expression 4s. The area of a square is found using
the expression s 2 . Complete the table.
b. Graph the values from the table on the same coordinate plane. Side length is represented on the x- axis and both perimeter and area are represented on the y-axis.
c. Sample answer: The graph representing perimeter of a square is a line because each side length is multiplied by 4. The graph representing area of a square is nonlinear because each side length is squared. Squared values do not increase at a constant rate.
28. To find the volume of a cube, multiply its base, its height, and its width.
What is the volume of the cube expressed as a power?
A. 6 2 in
3
SOLUTION: A cube has the same measures for all sides, so the base, height, and width are all 6 inches. The volume
is 6 • 6 • 6 or 6 3 inches cubed. This corresponds to
choice B.
29. Short Response The volume of an ice cube in
cubic millimeters is represented by the term 11 3 .
What is 11 3 in standard form?
SOLUTION:
30. What is the value of x2 – y 4 if x = –3 and y = –2?
F. –7 G. –2 H. 2 I. 7
SOLUTION:
Replace x with –3 and y with –2 in the expression
x2 – y 4. Then write the powers as products and simplify.
The correct answer is F.
31. The table below shows the number of ants in an ant farm on different days. The number of ants doubles every ten days.
a. How many ants were in the farm on Day 1? b. How many ants will be in the farm on Day 91?
SOLUTION: a. Use the look for a pattern strategy. The number of ants doubles every 10 days, so going backward, the number of ants is halved each time the number of days is decreased by 10. Extend the pattern to find the number of ants on Day 1.
So, there were 10 ants in the farm on Day 1. b. Extend the pattern to find the number of ants in the farm on Day 91. The number of ants doubles every 10 days.
So, there will be 5,120 ants in the farm on Day 91.
Day 61 51 41 31 21 11 1 Number of ants
640 320 160 80 40 20 10
Day 51 61 71 81 91 Number of ants
320 640 1,280 2,560 5,120
32. Nieves and her three friends are playing a video game. The table shows their scores at the end of the first round.
a. What is the difference between the highest and lowest scores? b. By how many points is Nieves losing to Polly?
SOLUTION: a. The highest score is 230 points. The lowest score is –189. Subtract: 230 – (–189) = 230 + 189 or 419 The difference between the highest and lowest scores is 419 points. b. Nieves' score is –189. Polly's score is –142. Subtract: –142 – (–189) = –142 + 189 or 47 Polly is ahead of Nieves by 47 points.
Add. 33. –12 + (–19)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –12 + (–19) = –31
34. –8 + (–11)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –8 + (–11) = –19
35. –5 + 6
SOLUTION: Since the signs are different, subtract the absolute values of the numbers. Since the largest absolute value is 6, the sign of the answer is the same as the sign of the 6. –5 + 6 = 1
Write the expression using exponents. 1. (–5)(–5)(–5)(–5)
SOLUTION: The base –5 is a factor 4 times, so the exponent is 4.
(–5)(–5)(–5)(–5) = (–5) 4
2. 3 • 3 • 5 • q • q • q
SOLUTION: The base 3 is a factor 2 times, so its exponent is 2. The base 5 is a factor 1 time, so its exponent is 1. Because a number without an exponent is the same as a number with an exponent of 1, write 5 without an exponent. The base q is a factor 3 times, so its exponent is 3.
3 • 3 • 5 • q • q • q = 3 2 • 5 • q3
3. m • m • m • m • m
SOLUTION: The base m is a factor 5 times, so its exponent is 5.
m • m • m • m • m = m 5
Evaluate the expression.
4. (–9) 4
(–9) 4 = (–9) • (–9) • (–9) • (–9) or 6,561
5.
or
6.
7. In the United States, nearly 8 • 10 9 text messages
are sent every month. About how many text messages is this?
SOLUTION: Write the power as a product and multiply.
There are about 8 billion text messages sent in the United States each month.
8. Interstate 70 stretches almost 2 3 • 5
2 • 11 miles
across the United States. About how many miles long is Interstate 70?
SOLUTION: Write the power as a product and multiply.
Interstate 70 stretches for almost 2,200 miles.
Evaluate the expression.
9. g5 – h 3 if g = 2 and h = 7
SOLUTION: Replace g with 2 and h with 7 in the expression
. Then write the powers as products and simplify.
10. c 2 + d
3 if c = 8 and d = –3
SOLUTION: Replace c with 8 and d with –3 in the expression
. Then write the powers as products and simplify.
11. a 2 • b
SOLUTION:
Replace a with and b with 2 in the expression
. Then write the powers as products and simplify.
12. (r – s) 3 + r
2 if r = –3 and s = –4
SOLUTION: Replace r with –3 and s with –4 in the expression
. Then write the powers as products and simplify.
13. Model with Mathematics Refer to the graphic novel frame below for Exercises a–c.
The metric system is based on powers of 10. For example, one kilometer is equal to 1,000 meters or
10 3 meters. Write each measurement in meters as a
power of 10. a. megameter (1,000,000 meters) b. gigameter (1,000,000,000 meters) c. petameter (1,000,000,000,000,000 meters)
SOLUTION: a. megameter: 1,000,000 meters has 6 zeros or
powers of 10. So, a megameter is 10 6 .
b. gigameter: 1,000,000,000 meters has 9 zeros or
powers of 10. So, a gigameter is 10 9 .
c. petameter: 1,000,000,000,000,000 meters has 15
.
14. Identify Structure Write an expression with an exponent that has a value between 0 and 1.
SOLUTION:
3 = 27, 3
2 = 9, 3
1 = 3.
.
1 = 3
To go from 81 to 27, divide 81 by 3. This is the same for the rest of the values: 27 ÷ 3 = 9, and 9 ÷ 3 = 3.
To find the value of 2 -1
, follow the same pattern.
2 3 = 8, 2
0 = 2 ÷ 2 or
2 3 • 3
4
A. 3 • 3 • 4 • 4 • 4 B. 2 • 2 • 2 • 3 • 3 • 3 C. 2 • 2 • 2 • 3 • 3 • 3 • 3 D. 6 • 12
SOLUTION:
2 3 • 34
= 2 • 2 • 2 • 3 • 3 • 3 • 3. This corresponds to choice C.
17. Write 3 · p p p · 3 · 3 using exponents.
18. Evaluate x3 + y 4 if x = −3 and y = 4.
SOLUTION: Replace x with –3 and y with 4 in the expression
. Then write the powers as products and simplify.
Write the expression using exponents.
19.
SOLUTION:
The base is a factor 3 times, so its exponent is
3.
20. s • (–7) • s • (–7) • (–7)
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base –7 is a factor 3 times, so its exponent is 3. The base s is a factor 2 times, so its exponent is 2.
21. 4 • b • b • 4 • b • b
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base 4 is a factor 2 times, so its exponent is 2. The base b is a factor 4 times, so its exponent is 4.
Evaluate the expression.
SOLUTION:
Replace k with 3 and m with in the expression
k 4 • m. Then write the powers as products and
simplify.
3 , if c = –1 and d = 2
SOLUTION: Replace c with –1 and d with 2 in the expression
. Then write the powers as
products and simplify.
statement.
2
SOLUTION: Simplify each expression.
28 is larger than 25, so > will make the statement true.
(6 – 2) 2 + 3 • 4 > 5
2
3 3
SOLUTION: Simplify each expression.
81 is equal to 81, so = will make the statement true.
5 + 7 2 + 3
true.
=
27. Multiple Representations A square has a side length of s inches. a. Tables Copy and complete the table showing the side length, perimeter, and area of the square on a separate piece of paper.
b. Graphs On a separate piece of grid paper, graph the ordered pairs (side length, perimeter) and (side length, area) on the same coordinate plane. Then connect the points for each set. c. Words On a separate piece of paper, compare and contrast the graphs of the perimeter and area of the square. Which graph is a line?
SOLUTION: a. The perimeter of a square is found using the expression 4s. The area of a square is found using
the expression s 2 . Complete the table.
b. Graph the values from the table on the same coordinate plane. Side length is represented on the x- axis and both perimeter and area are represented on the y-axis.
c. Sample answer: The graph representing perimeter of a square is a line because each side length is multiplied by 4. The graph representing area of a square is nonlinear because each side length is squared. Squared values do not increase at a constant rate.
28. To find the volume of a cube, multiply its base, its height, and its width.
What is the volume of the cube expressed as a power?
A. 6 2 in
3
SOLUTION: A cube has the same measures for all sides, so the base, height, and width are all 6 inches. The volume
is 6 • 6 • 6 or 6 3 inches cubed. This corresponds to
choice B.
29. Short Response The volume of an ice cube in
cubic millimeters is represented by the term 11 3 .
What is 11 3 in standard form?
SOLUTION:
30. What is the value of x2 – y 4 if x = –3 and y = –2?
F. –7 G. –2 H. 2 I. 7
SOLUTION:
Replace x with –3 and y with –2 in the expression
x2 – y 4. Then write the powers as products and simplify.
The correct answer is F.
31. The table below shows the number of ants in an ant farm on different days. The number of ants doubles every ten days.
a. How many ants were in the farm on Day 1? b. How many ants will be in the farm on Day 91?
SOLUTION: a. Use the look for a pattern strategy. The number of ants doubles every 10 days, so going backward, the number of ants is halved each time the number of days is decreased by 10. Extend the pattern to find the number of ants on Day 1.
So, there were 10 ants in the farm on Day 1. b. Extend the pattern to find the number of ants in the farm on Day 91. The number of ants doubles every 10 days.
So, there will be 5,120 ants in the farm on Day 91.
Day 61 51 41 31 21 11 1 Number of ants
640 320 160 80 40 20 10
Day 51 61 71 81 91 Number of ants
320 640 1,280 2,560 5,120
32. Nieves and her three friends are playing a video game. The table shows their scores at the end of the first round.
a. What is the difference between the highest and lowest scores? b. By how many points is Nieves losing to Polly?
SOLUTION: a. The highest score is 230 points. The lowest score is –189. Subtract: 230 – (–189) = 230 + 189 or 419 The difference between the highest and lowest scores is 419 points. b. Nieves' score is –189. Polly's score is –142. Subtract: –142 – (–189) = –142 + 189 or 47 Polly is ahead of Nieves by 47 points.
Add. 33. –12 + (–19)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –12 + (–19) = –31
34. –8 + (–11)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –8 + (–11) = –19
35. –5 + 6
SOLUTION: Since the signs
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41
Write the expression using exponents. 1. (–5)(–5)(–5)(–5)
SOLUTION: The base –5 is a factor 4 times, so the exponent is 4.
(–5)(–5)(–5)(–5) = (–5) 4
2. 3 • 3 • 5 • q • q • q
SOLUTION: The base 3 is a factor 2 times, so its exponent is 2. The base 5 is a factor 1 time, so its exponent is 1. Because a number without an exponent is the same as a number with an exponent of 1, write 5 without an exponent. The base q is a factor 3 times, so its exponent is 3.
3 • 3 • 5 • q • q • q = 3 2 • 5 • q3
3. m • m • m • m • m
SOLUTION: The base m is a factor 5 times, so its exponent is 5.
m • m • m • m • m = m 5
Evaluate the expression.
4. (–9) 4
(–9) 4 = (–9) • (–9) • (–9) • (–9) or 6,561
5.
or
6.
7. In the United States, nearly 8 • 10 9 text messages
are sent every month. About how many text messages is this?
SOLUTION: Write the power as a product and multiply.
There are about 8 billion text messages sent in the United States each month.
8. Interstate 70 stretches almost 2 3 • 5
2 • 11 miles
across the United States. About how many miles long is Interstate 70?
SOLUTION: Write the power as a product and multiply.
Interstate 70 stretches for almost 2,200 miles.
Evaluate the expression.
9. g5 – h 3 if g = 2 and h = 7
SOLUTION: Replace g with 2 and h with 7 in the expression
. Then write the powers as products and simplify.
10. c 2 + d
3 if c = 8 and d = –3
SOLUTION: Replace c with 8 and d with –3 in the expression
. Then write the powers as products and simplify.
11. a 2 • b
SOLUTION:
Replace a with and b with 2 in the expression
. Then write the powers as products and simplify.
12. (r – s) 3 + r
2 if r = –3 and s = –4
SOLUTION: Replace r with –3 and s with –4 in the expression
. Then write the powers as products and simplify.
13. Model with Mathematics Refer to the graphic novel frame below for Exercises a–c.
The metric system is based on powers of 10. For example, one kilometer is equal to 1,000 meters or
10 3 meters. Write each measurement in meters as a
power of 10. a. megameter (1,000,000 meters) b. gigameter (1,000,000,000 meters) c. petameter (1,000,000,000,000,000 meters)
SOLUTION: a. megameter: 1,000,000 meters has 6 zeros or
powers of 10. So, a megameter is 10 6 .
b. gigameter: 1,000,000,000 meters has 9 zeros or
powers of 10. So, a gigameter is 10 9 .
c. petameter: 1,000,000,000,000,000 meters has 15
.
14. Identify Structure Write an expression with an exponent that has a value between 0 and 1.
SOLUTION:
3 = 27, 3
2 = 9, 3
1 = 3.
.
1 = 3
To go from 81 to 27, divide 81 by 3. This is the same for the rest of the values: 27 ÷ 3 = 9, and 9 ÷ 3 = 3.
To find the value of 2 -1
, follow the same pattern.
2 3 = 8, 2
0 = 2 ÷ 2 or
2 3 • 3
4
A. 3 • 3 • 4 • 4 • 4 B. 2 • 2 • 2 • 3 • 3 • 3 C. 2 • 2 • 2 • 3 • 3 • 3 • 3 D. 6 • 12
SOLUTION:
2 3 • 34
= 2 • 2 • 2 • 3 • 3 • 3 • 3. This corresponds to choice C.
17. Write 3 · p p p · 3 · 3 using exponents.
18. Evaluate x3 + y 4 if x = −3 and y = 4.
SOLUTION: Replace x with –3 and y with 4 in the expression
. Then write the powers as products and simplify.
Write the expression using exponents.
19.
SOLUTION:
The base is a factor 3 times, so its exponent is
3.
20. s • (–7) • s • (–7) • (–7)
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base –7 is a factor 3 times, so its exponent is 3. The base s is a factor 2 times, so its exponent is 2.
21. 4 • b • b • 4 • b • b
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base 4 is a factor 2 times, so its exponent is 2. The base b is a factor 4 times, so its exponent is 4.
Evaluate the expression.
SOLUTION:
Replace k with 3 and m with in the expression
k 4 • m. Then write the powers as products and
simplify.
3 , if c = –1 and d = 2
SOLUTION: Replace c with –1 and d with 2 in the expression
. Then write the powers as
products and simplify.
statement.
2
SOLUTION: Simplify each expression.
28 is larger than 25, so > will make the statement true.
(6 – 2) 2 + 3 • 4 > 5
2
3 3
SOLUTION: Simplify each expression.
81 is equal to 81, so = will make the statement true.
5 + 7 2 + 3
true.
=
27. Multiple Representations A square has a side length of s inches. a. Tables Copy and complete the table showing the side length, perimeter, and area of the square on a separate piece of paper.
b. Graphs On a separate piece of grid paper, graph the ordered pairs (side length, perimeter) and (side length, area) on the same coordinate plane. Then connect the points for each set. c. Words On a separate piece of paper, compare and contrast the graphs of the perimeter and area of the square. Which graph is a line?
SOLUTION: a. The perimeter of a square is found using the expression 4s. The area of a square is found using
the expression s 2 . Complete the table.
b. Graph the values from the table on the same coordinate plane. Side length is represented on the x- axis and both perimeter and area are represented on the y-axis.
c. Sample answer: The graph representing perimeter of a square is a line because each side length is multiplied by 4. The graph representing area of a square is nonlinear because each side length is squared. Squared values do not increase at a constant rate.
28. To find the volume of a cube, multiply its base, its height, and its width.
What is the volume of the cube expressed as a power?
A. 6 2 in
3
SOLUTION: A cube has the same measures for all sides, so the base, height, and width are all 6 inches. The volume
is 6 • 6 • 6 or 6 3 inches cubed. This corresponds to
choice B.
29. Short Response The volume of an ice cube in
cubic millimeters is represented by the term 11 3 .
What is 11 3 in standard form?
SOLUTION:
30. What is the value of x2 – y 4 if x = –3 and y = –2?
F. –7 G. –2 H. 2 I. 7
SOLUTION:
Replace x with –3 and y with –2 in the expression
x2 – y 4. Then write the powers as products and simplify.
The correct answer is F.
31. The table below shows the number of ants in an ant farm on different days. The number of ants doubles every ten days.
a. How many ants were in the farm on Day 1? b. How many ants will be in the farm on Day 91?
SOLUTION: a. Use the look for a pattern strategy. The number of ants doubles every 10 days, so going backward, the number of ants is halved each time the number of days is decreased by 10. Extend the pattern to find the number of ants on Day 1.
So, there were 10 ants in the farm on Day 1. b. Extend the pattern to find the number of ants in the farm on Day 91. The number of ants doubles every 10 days.
So, there will be 5,120 ants in the farm on Day 91.
Day 61 51 41 31 21 11 1 Number of ants
640 320 160 80 40 20 10
Day 51 61 71 81 91 Number of ants
320 640 1,280 2,560 5,120
32. Nieves and her three friends are playing a video game. The table shows their scores at the end of the first round.
a. What is the difference between the highest and lowest scores? b. By how many points is Nieves losing to Polly?
SOLUTION: a. The highest score is 230 points. The lowest score is –189. Subtract: 230 – (–189) = 230 + 189 or 419 The difference between the highest and lowest scores is 419 points. b. Nieves' score is –189. Polly's score is –142. Subtract: –142 – (–189) = –142 + 189 or 47 Polly is ahead of Nieves by 47 points.
Add. 33. –12 + (–19)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –12 + (–19) = –31
34. –8 + (–11)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –8 + (–11) = –19
35. –5 + 6
SOLUTION: Since the signs are different, subtract the absolute values of the numbers. Since the largest absolute value is 6, the sign of the answer is the same as the sign of the 6. –5 + 6 = 1
Write the expression using exponents. 1. (–5)(–5)(–5)(–5)
SOLUTION: The base –5 is a factor 4 times, so the exponent is 4.
(–5)(–5)(–5)(–5) = (–5) 4
2. 3 • 3 • 5 • q • q • q
SOLUTION: The base 3 is a factor 2 times, so its exponent is 2. The base 5 is a factor 1 time, so its exponent is 1. Because a number without an exponent is the same as a number with an exponent of 1, write 5 without an exponent. The base q is a factor 3 times, so its exponent is 3.
3 • 3 • 5 • q • q • q = 3 2 • 5 • q3
3. m • m • m • m • m
SOLUTION: The base m is a factor 5 times, so its exponent is 5.
m • m • m • m • m = m 5
Evaluate the expression.
4. (–9) 4
(–9) 4 = (–9) • (–9) • (–9) • (–9) or 6,561
5.
or
6.
7. In the United States, nearly 8 • 10 9 text messages
are sent every month. About how many text messages is this?
SOLUTION: Write the power as a product and multiply.
There are about 8 billion text messages sent in the United States each month.
8. Interstate 70 stretches almost 2 3 • 5
2 • 11 miles
across the United States. About how many miles long is Interstate 70?
SOLUTION: Write the power as a product and multiply.
Interstate 70 stretches for almost 2,200 miles.
Evaluate the expression.
9. g5 – h 3 if g = 2 and h = 7
SOLUTION: Replace g with 2 and h with 7 in the expression
. Then write the powers as products and simplify.
10. c 2 + d
3 if c = 8 and d = –3
SOLUTION: Replace c with 8 and d with –3 in the expression
. Then write the powers as products and simplify.
11. a 2 • b
SOLUTION:
Replace a with and b with 2 in the expression
. Then write the powers as products and simplify.
12. (r – s) 3 + r
2 if r = –3 and s = –4
SOLUTION: Replace r with –3 and s with –4 in the expression
. Then write the powers as products and simplify.
13. Model with Mathematics Refer to the graphic novel frame below for Exercises a–c.
The metric system is based on powers of 10. For example, one kilometer is equal to 1,000 meters or
10 3 meters. Write each measurement in meters as a
power of 10. a. megameter (1,000,000 meters) b. gigameter (1,000,000,000 meters) c. petameter (1,000,000,000,000,000 meters)
SOLUTION: a. megameter: 1,000,000 meters has 6 zeros or
powers of 10. So, a megameter is 10 6 .
b. gigameter: 1,000,000,000 meters has 9 zeros or
powers of 10. So, a gigameter is 10 9 .
c. petameter: 1,000,000,000,000,000 meters has 15
.
14. Identify Structure Write an expression with an exponent that has a value between 0 and 1.
SOLUTION:
3 = 27, 3
2 = 9, 3
1 = 3.
.
1 = 3
To go from 81 to 27, divide 81 by 3. This is the same for the rest of the values: 27 ÷ 3 = 9, and 9 ÷ 3 = 3.
To find the value of 2 -1
, follow the same pattern.
2 3 = 8, 2
0 = 2 ÷ 2 or
2 3 • 3
4
A. 3 • 3 • 4 • 4 • 4 B. 2 • 2 • 2 • 3 • 3 • 3 C. 2 • 2 • 2 • 3 • 3 • 3 • 3 D. 6 • 12
SOLUTION:
2 3 • 34
= 2 • 2 • 2 • 3 • 3 • 3 • 3. This corresponds to choice C.
17. Write 3 · p p p · 3 · 3 using exponents.
18. Evaluate x3 + y 4 if x = −3 and y = 4.
SOLUTION: Replace x with –3 and y with 4 in the expression
. Then write the powers as products and simplify.
Write the expression using exponents.
19.
SOLUTION:
The base is a factor 3 times, so its exponent is
3.
20. s • (–7) • s • (–7) • (–7)
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base –7 is a factor 3 times, so its exponent is 3. The base s is a factor 2 times, so its exponent is 2.
21. 4 • b • b • 4 • b • b
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base 4 is a factor 2 times, so its exponent is 2. The base b is a factor 4 times, so its exponent is 4.
Evaluate the expression.
SOLUTION:
Replace k with 3 and m with in the expression
k 4 • m. Then write the powers as products and
simplify.
3 , if c = –1 and d = 2
SOLUTION: Replace c with –1 and d with 2 in the expression
. Then write the powers as
products and simplify.
statement.
2
SOLUTION: Simplify each expression.
28 is larger than 25, so > will make the statement true.
(6 – 2) 2 + 3 • 4 > 5
2
3 3
SOLUTION: Simplify each expression.
81 is equal to 81, so = will make the statement true.
5 + 7 2 + 3
true.
=
27. Multiple Representations A square has a side length of s inches. a. Tables Copy and complete the table showing the side length, perimeter, and area of the square on a separate piece of paper.
b. Graphs On a separate piece of grid paper, graph the ordered pairs (side length, perimeter) and (side length, area) on the same coordinate plane. Then connect the points for each set. c. Words On a separate piece of paper, compare and contrast the graphs of the perimeter and area of the square. Which graph is a line?
SOLUTION: a. The perimeter of a square is found using the expression 4s. The area of a square is found using
the expression s 2 . Complete the table.
b. Graph the values from the table on the same coordinate plane. Side length is represented on the x- axis and both perimeter and area are represented on the y-axis.
c. Sample answer: The graph representing perimeter of a square is a line because each side length is multiplied by 4. The graph representing area of a square is nonlinear because each side length is squared. Squared values do not increase at a constant rate.
28. To find the volume of a cube, multiply its base, its height, and its width.
What is the volume of the cube expressed as a power?
A. 6 2 in
3
SOLUTION: A cube has the same measures for all sides, so the base, height, and width are all 6 inches. The volume
is 6 • 6 • 6 or 6 3 inches cubed. This corresponds to
choice B.
29. Short Response The volume of an ice cube in
cubic millimeters is represented by the term 11 3 .
What is 11 3 in standard form?
SOLUTION:
30. What is the value of x2 – y 4 if x = –3 and y = –2?
F. –7 G. –2 H. 2 I. 7
SOLUTION:
Replace x with –3 and y with –2 in the expression
x2 – y 4. Then write the powers as products and simplify.
The correct answer is F.
31. The table below shows the number of ants in an ant farm on different days. The number of ants doubles every ten days.
a. How many ants were in the farm on Day 1? b. How many ants will be in the farm on Day 91?
SOLUTION: a. Use the look for a pattern strategy. The number of ants doubles every 10 days, so going backward, the number of ants is halved each time the number of days is decreased by 10. Extend the pattern to find the number of ants on Day 1.
So, there were 10 ants in the farm on Day 1. b. Extend the pattern to find the number of ants in the farm on Day 91. The number of ants doubles every 10 days.
So, there will be 5,120 ants in the farm on Day 91.
Day 61 51 41 31 21 11 1 Number of ants
640 320 160 80 40 20 10
Day 51 61 71 81 91 Number of ants
320 640 1,280 2,560 5,120
32. Nieves and her three friends are playing a video game. The table shows their scores at the end of the first round.
a. What is the difference between the highest and lowest scores? b. By how many points is Nieves losing to Polly?
SOLUTION: a. The highest score is 230 points. The lowest score is –189. Subtract: 230 – (–189) = 230 + 189 or 419 The difference between the highest and lowest scores is 419 points. b. Nieves' score is –189. Polly's score is –142. Subtract: –142 – (–189) = –142 + 189 or 47 Polly is ahead of Nieves by 47 points.
Add. 33. –12 + (–19)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –12 + (–19) = –31
34. –8 + (–11)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –8 + (–11) = –19
35. –5 + 6
SOLUTION: Since the signs are different, subtract the absolute values of the numbers. Since the largest absolute value is 6, the sign of the answer is the same as the sign of the 6. –5 + 6 = 1
eSolutions Manual - Powered by Cognero Page 1
1-2 Powers and Exponents
Write the expression using exponents. 1. (–5)(–5)(–5)(–5)
SOLUTION: The base –5 is a factor 4 times, so the exponent is 4.
(–5)(–5)(–5)(–5) = (–5) 4
2. 3 • 3 • 5 • q • q • q
SOLUTION: The base 3 is a factor 2 times, so its exponent is 2. The base 5 is a factor 1 time, so its exponent is 1. Because a number without an exponent is the same as a number with an exponent of 1, write 5 without an exponent. The base q is a factor 3 times, so its exponent is 3.
3 • 3 • 5 • q • q • q = 3 2 • 5 • q3
3. m • m • m • m • m
SOLUTION: The base m is a factor 5 times, so its exponent is 5.
m • m • m • m • m = m 5
Evaluate the expression.
4. (–9) 4
(–9) 4 = (–9) • (–9) • (–9) • (–9) or 6,561
5.
or
6.
7. In the United States, nearly 8 • 10 9 text messages
are sent every month. About how many text messages is this?
SOLUTION: Write the power as a product and multiply.
There are about 8 billion text messages sent in the United States each month.
8. Interstate 70 stretches almost 2 3 • 5
2 • 11 miles
across the United States. About how many miles long is Interstate 70?
SOLUTION: Write the power as a product and multiply.
Interstate 70 stretches for almost 2,200 miles.
Evaluate the expression.
9. g5 – h 3 if g = 2 and h = 7
SOLUTION: Replace g with 2 and h with 7 in the expression
. Then write the powers as products and simplify.
10. c 2 + d
3 if c = 8 and d = –3
SOLUTION: Replace c with 8 and d with –3 in the expression
. Then write the powers as products and simplify.
11. a 2 • b
SOLUTION:
Replace a with and b with 2 in the expression
. Then write the powers as products and simplify.
12. (r – s) 3 + r
2 if r = –3 and s = –4
SOLUTION: Replace r with –3 and s with –4 in the expression
. Then write the powers as products and simplify.
13. Model with Mathematics Refer to the graphic novel frame below for Exercises a–c.
The metric system is based on powers of 10. For example, one kilometer is equal to 1,000 meters or
10 3 meters. Write each measurement in meters as a
power of 10. a. megameter (1,000,000 meters) b. gigameter (1,000,000,000 meters) c. petameter (1,000,000,000,000,000 meters)
SOLUTION: a. megameter: 1,000,000 meters has 6 zeros or
powers of 10. So, a megameter is 10 6 .
b. gigameter: 1,000,000,000 meters has 9 zeros or
powers of 10. So, a gigameter is 10 9 .
c. petameter: 1,000,000,000,000,000 meters has 15
.
14. Identify Structure Write an expression with an exponent that has a value between 0 and 1.
SOLUTION:
3 = 27, 3
2 = 9, 3
1 = 3.
.
1 = 3
To go from 81 to 27, divide 81 by 3. This is the same for the rest of the values: 27 ÷ 3 = 9, and 9 ÷ 3 = 3.
To find the value of 2 -1
, follow the same pattern.
2 3 = 8, 2
0 = 2 ÷ 2 or
2 3 • 3
4
A. 3 • 3 • 4 • 4 • 4 B. 2 • 2 • 2 • 3 • 3 • 3 C. 2 • 2 • 2 • 3 • 3 • 3 • 3 D. 6 • 12
SOLUTION:
2 3 • 34
= 2 • 2 • 2 • 3 • 3 • 3 • 3. This corresponds to choice C.
17. Write 3 · p p p · 3 · 3 using exponents.
18. Evaluate x3 + y 4 if x = −3 and y = 4.
SOLUTION: Replace x with –3 and y with 4 in the expression
. Then write the powers as products and simplify.
Write the expression using exponents.
19.
SOLUTION:
The base is a factor 3 times, so its exponent is
3.
20. s • (–7) • s • (–7) • (–7)
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base –7 is a factor 3 times, so its exponent is 3. The base s is a factor 2 times, so its exponent is 2.
21. 4 • b • b • 4 • b • b
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base 4 is a factor 2 times, so its exponent is 2. The base b is a factor 4 times, so its exponent is 4.
Evaluate the expression.
SOLUTION:
Replace k with 3 and m with in the expression
k 4 • m. Then write the powers as products and
simplify.
3 , if c = –1 and d = 2
SOLUTION: Replace c with –1 and d with 2 in the expression
. Then write the powers as
products and simplify.
statement.
2
SOLUTION: Simplify each expression.
28 is larger than 25, so > will make the statement true.
(6 – 2) 2 + 3 • 4 > 5
2
3 3
SOLUTION: Simplify each expression.
81 is equal to 81, so = will make the statement true.
5 + 7 2 + 3
true.
=
27. Multiple Representations A square has a side length of s inches. a. Tables Copy and complete the table showing the side length, perimeter, and area of the square on a separate piece of paper.
b. Graphs On a separate piece of grid paper, graph the ordered pairs (side length, perimeter) and (side length, area) on the same coordinate plane. Then connect the points for each set. c. Words On a separate piece of paper, compare and contrast the graphs of the perimeter and area of the square. Which graph is a line?
SOLUTION: a. The perimeter of a square is found using the expression 4s. The area of a square is found using
the expression s 2 . Complete the table.
b. Graph the values from the table on the same coordinate plane. Side length is represented on the x- axis and both perimeter and area are represented on the y-axis.
c. Sample answer: The graph representing perimeter of a square is a line because each side length is multiplied by 4. The graph representing area of a square is nonlinear because each side length is squared. Squared values do not increase at a constant rate.
28. To find the volume of a cube, multiply its base, its height, and its width.
What is the volume of the cube expressed as a power?
A. 6 2 in
3
SOLUTION: A cube has the same measures for all sides, so the base, height, and width are all 6 inches. The volume
is 6 • 6 • 6 or 6 3 inches cubed. This corresponds to
choice B.
29. Short Response The volume of an ice cube in
cubic millimeters is represented by the term 11 3 .
What is 11 3 in standard form?
SOLUTION:
30. What is the value of x2 – y 4 if x = –3 and y = –2?
F. –7 G. –2 H. 2 I. 7
SOLUTION:
Replace x with –3 and y with –2 in the expression
x2 – y 4. Then write the powers as products and simplify.
The correct answer is F.
31. The table below shows the number of ants in an ant farm on different days. The number of ants doubles every ten days.
a. How many ants were in the farm on Day 1? b. How many ants will be in the farm on Day 91?
SOLUTION: a. Use the look for a pattern strategy. The number of ants doubles every 10 days, so going backward, the number of ants is halved each time the number of days is decreased by 10. Extend the pattern to find the number of ants on Day 1.
So, there were 10 ants in the farm on Day 1. b. Extend the pattern to find the number of ants in the farm on Day 91. The number of ants doubles every 10 days.
So, there will be 5,120 ants in the farm on Day 91.
Day 61 51 41 31 21 11 1 Number of ants
640 320 160 80 40 20 10
Day 51 61 71 81 91 Number of ants
320 640 1,280 2,560 5,120
32. Nieves and her three friends are playing a video game. The table shows their scores at the end of the first round.
a. What is the difference between the highest and lowest scores? b. By how many points is Nieves losing to Polly?
SOLUTION: a. The highest score is 230 points. The lowest score is –189. Subtract: 230 – (–189) = 230 + 189 or 419 The difference between the highest and lowest scores is 419 points. b. Nieves' score is –189. Polly's score is –142. Subtract: –142 – (–189) = –142 + 189 or 47 Polly is ahead of Nieves by 47 points.
Add. 33. –12 + (–19)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –12 + (–19) = –31
34. –8 + (–11)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –8 + (–11) = –19
35. –5 + 6
SOLUTION: Since the signs are different, subtract the absolute values of the numbers. Since the largest absolute value is 6, the sign of the answer is the same as the sign of the 6. –5 + 6 = 1
Write the expression using exponents. 1. (–5)(–5)(–5)(–5)
SOLUTION: The base –5 is a factor 4 times, so the exponent is 4.
(–5)(–5)(–5)(–5) = (–5) 4
2. 3 • 3 • 5 • q • q • q
SOLUTION: The base 3 is a factor 2 times, so its exponent is 2. The base 5 is a factor 1 time, so its exponent is 1. Because a number without an exponent is the same as a number with an exponent of 1, write 5 without an exponent. The base q is a factor 3 times, so its exponent is 3.
3 • 3 • 5 • q • q • q = 3 2 • 5 • q3
3. m • m • m • m • m
SOLUTION: The base m is a factor 5 times, so its exponent is 5.
m • m • m • m • m = m 5
Evaluate the expression.
4. (–9) 4
(–9) 4 = (–9) • (–9) • (–9) • (–9) or 6,561
5.
or
6.
7. In the United States, nearly 8 • 10 9 text messages
are sent every month. About how many text messages is this?
SOLUTION: Write the power as a product and multiply.
There are about 8 billion text messages sent in the United States each month.
8. Interstate 70 stretches almost 2 3 • 5
2 • 11 miles
across the United States. About how many miles long is Interstate 70?
SOLUTION: Write the power as a product and multiply.
Interstate 70 stretches for almost 2,200 miles.
Evaluate the expression.
9. g5 – h 3 if g = 2 and h = 7
SOLUTION: Replace g with 2 and h with 7 in the expression
. Then write the powers as products and simplify.
10. c 2 + d
3 if c = 8 and d = –3
SOLUTION: Replace c with 8 and d with –3 in the expression
. Then write the powers as products and simplify.
11. a 2 • b
SOLUTION:
Replace a with and b with 2 in the expression
. Then write the powers as products and simplify.
12. (r – s) 3 + r
2 if r = –3 and s = –4
SOLUTION: Replace r with –3 and s with –4 in the expression
. Then write the powers as products and simplify.
13. Model with Mathematics Refer to the graphic novel frame below for Exercises a–c.
The metric system is based on powers of 10. For example, one kilometer is equal to 1,000 meters or
10 3 meters. Write each measurement in meters as a
power of 10. a. megameter (1,000,000 meters) b. gigameter (1,000,000,000 meters) c. petameter (1,000,000,000,000,000 meters)
SOLUTION: a. megameter: 1,000,000 meters has 6 zeros or
powers of 10. So, a megameter is 10 6 .
b. gigameter: 1,000,000,000 meters has 9 zeros or
powers of 10. So, a gigameter is 10 9 .
c. petameter: 1,000,000,000,000,000 meters has 15
.
14. Identify Structure Write an expression with an exponent that has a value between 0 and 1.
SOLUTION:
3 = 27, 3
2 = 9, 3
1 = 3.
.
1 = 3
To go from 81 to 27, divide 81 by 3. This is the same for the rest of the values: 27 ÷ 3 = 9, and 9 ÷ 3 = 3.
To find the value of 2 -1
, follow the same pattern.
2 3 = 8, 2
0 = 2 ÷ 2 or
2 3 • 3
4
A. 3 • 3 • 4 • 4 • 4 B. 2 • 2 • 2 • 3 • 3 • 3 C. 2 • 2 • 2 • 3 • 3 • 3 • 3 D. 6 • 12
SOLUTION:
2 3 • 34
= 2 • 2 • 2 • 3 • 3 • 3 • 3. This corresponds to choice C.
17. Write 3 · p p p · 3 · 3 using exponents.
18. Evaluate x3 + y 4 if x = −3 and y = 4.
SOLUTION: Replace x with –3 and y with 4 in the expression
. Then write the powers as products and simplify.
Write the expression using exponents.
19.
SOLUTION:
The base is a factor 3 times, so its exponent is
3.
20. s • (–7) • s • (–7) • (–7)
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base –7 is a factor 3 times, so its exponent is 3. The base s is a factor 2 times, so its exponent is 2.
21. 4 • b • b • 4 • b • b
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base 4 is a factor 2 times, so its exponent is 2. The base b is a factor 4 times, so its exponent is 4.
Evaluate the expression.
SOLUTION:
Replace k with 3 and m with in the expression
k 4 • m. Then write the powers as products and
simplify.
3 , if c = –1 and d = 2
SOLUTION: Replace c with –1 and d with 2 in the expression
. Then write the powers as
products and simplify.
statement.
2
SOLUTION: Simplify each expression.
28 is larger than 25, so > will make the statement true.
(6 – 2) 2 + 3 • 4 > 5
2
3 3
SOLUTION: Simplify each expression.
81 is equal to 81, so = will make the statement true.
5 + 7 2 + 3
true.
=
27. Multiple Representations A square has a side length of s inches. a. Tables Copy and complete the table showing the side length, perimeter, and area of the square on a separate piece of paper.
b. Graphs On a separate piece of grid paper, graph the ordered pairs (side length, perimeter) and (side length, area) on the same coordinate plane. Then connect the points for each set. c. Words On a separate piece of paper, compare and contrast the graphs of the perimeter and area of the square. Which graph is a line?
SOLUTION: a. The perimeter of a square is found using the expression 4s. The area of a square is found using
the expression s 2 . Complete the table.
b. Graph the values from the table on the same coordinate plane. Side length is represented on the x- axis and both perimeter and area are represented on the y-axis.
c. Sample answer: The graph representing perimeter of a square is a line because each side length is multiplied by 4. The graph representing area of a square is nonlinear because each side length is squared. Squared values do not increase at a constant rate.
28. To find the volume of a cube, multiply its base, its height, and its width.
What is the volume of the cube expressed as a power?
A. 6 2 in
3
SOLUTION: A cube has the same measures for all sides, so the base, height, and width are all 6 inches. The volume
is 6 • 6 • 6 or 6 3 inches cubed. This corresponds to
choice B.
29. Short Response The volume of an ice cube in
cubic millimeters is represented by the term 11 3 .
What is 11 3 in standard form?
SOLUTION:
30. What is the value of x2 – y 4 if x = –3 and y = –2?
F. –7 G. –2 H. 2 I. 7
SOLUTION:
Replace x with –3 and y with –2 in the expression
x2 – y 4. Then write the powers as products and simplify.
The correct answer is F.
31. The table below shows the number of ants in an ant farm on different days. The number of ants doubles every ten days.
a. How many ants were in the farm on Day 1? b. How many ants will be in the farm on Day 91?
SOLUTION: a. Use the look for a pattern strategy. The number of ants doubles every 10 days, so going backward, the number of ants is halved each time the number of days is decreased by 10. Extend the pattern to find the number of ants on Day 1.
So, there were 10 ants in the farm on Day 1. b. Extend the pattern to find the number of ants in the farm on Day 91. The number of ants doubles every 10 days.
So, there will be 5,120 ants in the farm on Day 91.
Day 61 51 41 31 21 11 1 Number of ants
640 320 160 80 40 20 10
Day 51 61 71 81 91 Number of ants
320 640 1,280 2,560 5,120
32. Nieves and her three friends are playing a video game. The table shows their scores at the end of the first round.
a. What is the difference between the highest and lowest scores? b. By how many points is Nieves losing to Polly?
SOLUTION: a. The highest score is 230 points. The lowest score is –189. Subtract: 230 – (–189) = 230 + 189 or 419 The difference between the highest and lowest scores is 419 points. b. Nieves' score is –189. Polly's score is –142. Subtract: –142 – (–189) = –142 + 189 or 47 Polly is ahead of Nieves by 47 points.
Add. 33. –12 + (–19)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –12 + (–19) = –31
34. –8 + (–11)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –8 + (–11) = –19
35. –5 + 6
SOLUTION: Since the signs are different, subtract the absolute values of the numbers. Since the largest absolute value is 6, the sign of the answer is the same as the sign of the 6. –5 + 6 = 1
eSolutions Manual - Powered by Cognero Page 2
1-2 Powers and Exponents
Write the expression using exponents. 1. (–5)(–5)(–5)(–5)
SOLUTION: The base –5 is a factor 4 times, so the exponent is 4.
(–5)(–5)(–5)(–5) = (–5) 4
2. 3 • 3 • 5 • q • q • q
SOLUTION: The base 3 is a factor 2 times, so its exponent is 2. The base 5 is a factor 1 time, so its exponent is 1. Because a number without an exponent is the same as a number with an exponent of 1, write 5 without an exponent. The base q is a factor 3 times, so its exponent is 3.
3 • 3 • 5 • q • q • q = 3 2 • 5 • q3
3. m • m • m • m • m
SOLUTION: The base m is a factor 5 times, so its exponent is 5.
m • m • m • m • m = m 5
Evaluate the expression.
4. (–9) 4
(–9) 4 = (–9) • (–9) • (–9) • (–9) or 6,561
5.
or
6.
7. In the United States, nearly 8 • 10 9 text messages
are sent every month. About how many text messages is this?
SOLUTION: Write the power as a product and multiply.
There are about 8 billion text messages sent in the United States each month.
8. Interstate 70 stretches almost 2 3 • 5
2 • 11 miles
across the United States. About how many miles long is Interstate 70?
SOLUTION: Write the power as a product and multiply.
Interstate 70 stretches for almost 2,200 miles.
Evaluate the expression.
9. g5 – h 3 if g = 2 and h = 7
SOLUTION: Replace g with 2 and h with 7 in the expression
. Then write the powers as products and simplify.
10. c 2 + d
3 if c = 8 and d = –3
SOLUTION: Replace c with 8 and d with –3 in the expression
. Then write the powers as products and simplify.
11. a 2 • b
SOLUTION:
Replace a with and b with 2 in the expression
. Then write the powers as products and simplify.
12. (r – s) 3 + r
2 if r = –3 and s = –4
SOLUTION: Replace r with –3 and s with –4 in the expression
. Then write the powers as products and simplify.
13. Model with Mathematics Refer to the graphic novel frame below for Exercises a–c.
The metric system is based on powers of 10. For example, one kilometer is equal to 1,000 meters or
10 3 meters. Write each measurement in meters as a
power of 10. a. megameter (1,000,000 meters) b. gigameter (1,000,000,000 meters) c. petameter (1,000,000,000,000,000 meters)
SOLUTION: a. megameter: 1,000,000 meters has 6 zeros or
powers of 10. So, a megameter is 10 6 .
b. gigameter: 1,000,000,000 meters has 9 zeros or
powers of 10. So, a gigameter is 10 9 .
c. petameter: 1,000,000,000,000,000 meters has 15
.
14. Identify Structure Write an expression with an exponent that has a value between 0 and 1.
SOLUTION:
3 = 27, 3
2 = 9, 3
1 = 3.
.
1 = 3
To go from 81 to 27, divide 81 by 3. This is the same for the rest of the values: 27 ÷ 3 = 9, and 9 ÷ 3 = 3.
To find the value of 2 -1
, follow the same pattern.
2 3 = 8, 2
0 = 2 ÷ 2 or
2 3 • 3
4
A. 3 • 3 • 4 • 4 • 4 B. 2 • 2 • 2 • 3 • 3 • 3 C. 2 • 2 • 2 • 3 • 3 • 3 • 3 D. 6 • 12
SOLUTION:
2 3 • 34
= 2 • 2 • 2 • 3 • 3 • 3 • 3. This corresponds to choice C.
17. Write 3 · p p p · 3 · 3 using exponents.
18. Evaluate x3 + y 4 if x = −3 and y = 4.
SOLUTION: Replace x with –3 and y with 4 in the expression
. Then write the powers as products and simplify.
Write the expression using exponents.
19.
SOLUTION:
The base is a factor 3 times, so its exponent is
3.
20. s • (–7) • s • (–7) • (–7)
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base –7 is a factor 3 times, so its exponent is 3. The base s is a factor 2 times, so its exponent is 2.
21. 4 • b • b • 4 • b • b
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base 4 is a factor 2 times, so its exponent is 2. The base b is a factor 4 times, so its exponent is 4.
Evaluate the expression.
SOLUTION:
Replace k with 3 and m with in the expression
k 4 • m. Then write the powers as products and
simplify.
3 , if c = –1 and d = 2
SOLUTION: Replace c with –1 and d with 2 in the expression
. Then write the powers as
products and simplify.
statement.
2
SOLUTION: Simplify each expression.
28 is larger than 25, so > will make the statement true.
(6 – 2) 2 + 3 • 4 > 5
2
3 3
SOLUTION: Simplify each expression.
81 is equal to 81, so = will make the statement true.
5 + 7 2 + 3
true.
=
27. Multiple Representations A square has a side length of s inches. a. Tables Copy and complete the table showing the side length, perimeter, and area of the square on a separate piece of paper.
b. Graphs On a separate piece of grid paper, graph the ordered pairs (side length, perimeter) and (side length, area) on the same coordinate plane. Then connect the points for each set. c. Words On a separate piece of paper, compare and contrast the graphs of the perimeter and area of the square. Which graph is a line?
SOLUTION: a. The perimeter of a square is found using the expression 4s. The area of a square is found using
the expression s 2 . Complete the table.
b. Graph the values from the table on the same coordinate plane. Side length is represented on the x- axis and both perimeter and area are represented on the y-axis.
c. Sample answer: The graph representing perimeter of a square is a line because each side length is multiplied by 4. The graph representing area of a square is nonlinear because each side length is squared. Squared values do not increase at a constant rate.
28. To find the volume of a cube, multiply its base, its height, and its width.
What is the volume of the cube expressed as a power?
A. 6 2 in
3
SOLUTION: A cube has the same measures for all sides, so the base, height, and width are all 6 inches. The volume
is 6 • 6 • 6 or 6 3 inches cubed. This corresponds to
choice B.
29. Short Response The volume of an ice cube in
cubic millimeters is represented by the term 11 3 .
What is 11 3 in standard form?
SOLUTION:
30. What is the value of x2 – y 4 if x = –3 and y = –2?
F. –7 G. –2 H. 2 I. 7
SOLUTION:
Replace x with –3 and y with –2 in the expression
x2 – y 4. Then write the powers as products and simplify.
The correct answer is F.
31. The table below shows the number of ants in an ant farm on different days. The number of ants doubles every ten days.
a. How many ants were in the farm on Day 1? b. How many ants will be in the farm on Day 91?
SOLUTION: a. Use the look for a pattern strategy. The number of ants doubles every 10 days, so going backward, the number of ants is halved each time the number of days is decreased by 10. Extend the pattern to find the number of ants on Day 1.
So, there were 10 ants in the farm on Day 1. b. Extend the pattern to find the number of ants in the farm on Day 91. The number of ants doubles every 10 days.
So, there will be 5,120 ants in the farm on Day 91.
Day 61 51 41 31 21 11 1 Number of ants
640 320 160 80 40 20 10
Day 51 61 71 81 91 Number of ants
320 640 1,280 2,560 5,120
32. Nieves and her three friends are playing a video game. The table shows their scores at the end of the first round.
a. What is the difference between the highest and lowest scores? b. By how many points is Nieves losing to Polly?
SOLUTION: a. The highest score is 230 points. The lowest score is –189. Subtract: 230 – (–189) = 230 + 189 or 419 The difference between the highest and lowest scores is 419 points. b. Nieves' score is –189. Polly's score is –142. Subtract: –142 – (–189) = –142 + 189 or 47 Polly is ahead of Nieves by 47 points.
Add. 33. –12 + (–19)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –12 + (–19) = –31
34. –8 + (–11)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –8 + (–11) = –19
35. –5 + 6
SOLUTION: Since the signs are different, subtract the absolute values of the numbers. Since the largest absolute value is 6, the sign of the answer is the same as the sign of the 6. –5 + 6 = 1
Write the expression using exponents. 1. (–5)(–5)(–5)(–5)
SOLUTION: The base –5 is a factor 4 times, so the exponent is 4.
(–5)(–5)(–5)(–5) = (–5) 4
2. 3 • 3 • 5 • q • q • q
SOLUTION: The base 3 is a factor 2 times, so its exponent is 2. The base 5 is a factor 1 time, so its exponent is 1. Because a number without an exponent is the same as a number with an exponent of 1, write 5 without an exponent. The base q is a factor 3 times, so its exponent is 3.
3 • 3 • 5 • q • q • q = 3 2 • 5 • q3
3. m • m • m • m • m
SOLUTION: The base m is a factor 5 times, so its exponent is 5.
m • m • m • m • m = m 5
Evaluate the expression.
4. (–9) 4
(–9) 4 = (–9) • (–9) • (–9) • (–9) or 6,561
5.
or
6.
7. In the United States, nearly 8 • 10 9 text messages
are sent every month. About how many text messages is this?
SOLUTION: Write the power as a product and multiply.
There are about 8 billion text messages sent in the United States each month.
8. Interstate 70 stretches almost 2 3 • 5
2 • 11 miles
across the United States. About how many miles long is Interstate 70?
SOLUTION: Write the power as a product and multiply.
Interstate 70 stretches for almost 2,200 miles.
Evaluate the expression.
9. g5 – h 3 if g = 2 and h = 7
SOLUTION: Replace g with 2 and h with 7 in the expression
. Then write the powers as products and simplify.
10. c 2 + d
3 if c = 8 and d = –3
SOLUTION: Replace c with 8 and d with –3 in the expression
. Then write the powers as products and simplify.
11. a 2 • b
SOLUTION:
Replace a with and b with 2 in the expression
. Then write the powers as products and simplify.
12. (r – s) 3 + r
2 if r = –3 and s = –4
SOLUTION: Replace r with –3 and s with –4 in the expression
. Then write the powers as products and simplify.
13. Model with Mathematics Refer to the graphic novel frame below for Exercises a–c.
The metric system is based on powers of 10. For example, one kilometer is equal to 1,000 meters or
10 3 meters. Write each measurement in meters as a
power of 10. a. megameter (1,000,000 meters) b. gigameter (1,000,000,000 meters) c. petameter (1,000,000,000,000,000 meters)
SOLUTION: a. megameter: 1,000,000 meters has 6 zeros or
powers of 10. So, a megameter is 10 6 .
b. gigameter: 1,000,000,000 meters has 9 zeros or
powers of 10. So, a gigameter is 10 9 .
c. petameter: 1,000,000,000,000,000 meters has 15
.
14. Identify Structure Write an expression with an exponent that has a value between 0 and 1.
SOLUTION:
3 = 27, 3
2 = 9, 3
1 = 3.
.
1 = 3
To go from 81 to 27, divide 81 by 3. This is the same for the rest of the values: 27 ÷ 3 = 9, and 9 ÷ 3 = 3.
To find the value of 2 -1
, follow the same pattern.
2 3 = 8, 2
0 = 2 ÷ 2 or
2 3 • 3
4
A. 3 • 3 • 4 • 4 • 4 B. 2 • 2 • 2 • 3 • 3 • 3 C. 2 • 2 • 2 • 3 • 3 • 3 • 3 D. 6 • 12
SOLUTION:
2 3 • 34
= 2 • 2 • 2 • 3 • 3 • 3 • 3. This corresponds to choice C.
17. Write 3 · p p p · 3 · 3 using exponents.
18. Evaluate x3 + y 4 if x = −3 and y = 4.
SOLUTION: Replace x with –3 and y with 4 in the expression
. Then write the powers as products and simplify.
Write the expression using exponents.
19.
SOLUTION:
The base is a factor 3 times, so its exponent is
3.
20. s • (–7) • s • (–7) • (–7)
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base –7 is a factor 3 times, so its exponent is 3. The base s is a factor 2 times, so its exponent is 2.
21. 4 • b • b • 4 • b • b
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base 4 is a factor 2 times, so its exponent is 2. The base b is a factor 4 times, so its exponent is 4.
Evaluate the expression.
SOLUTION:
Replace k with 3 and m with in the expression
k 4 • m. Then write the powers as products and
simplify.
3 , if c = –1 and d = 2
SOLUTION: Replace c with –1 and d with 2 in the expression
. Then write the powers as
products and simplify.
statement.
2
SOLUTION: Simplify each expression.
28 is larger than 25, so > will make the statement true.
(6 – 2) 2 + 3 • 4 > 5
2
3 3
SOLUTION: Simplify each expression.
81 is equal to 81, so = will make the statement true.
5 + 7 2 + 3
true.
=
27. Multiple Representations A square has a side length of s inches. a. Tables Copy and complete the table showing the side length, perimeter, and area of the square on a separate piece of paper.
b. Graphs On a separate piece of grid paper, graph the ordered pairs (side length, perimeter) and (side length, area) on the same coordinate plane. Then connect the points for each set. c. Words On a separate piece of paper, compare and contrast the graphs of the perimeter and area of the square. Which graph is a line?
SOLUTION: a. The perimeter of a square is found using the expression 4s. The area of a square is found using
the expression s 2 . Complete the table.
b. Graph the values from the table on the same coordinate plane. Side length is represented on the x- axis and both perimeter and area are represented on the y-axis.
c. Sample answer: The graph representing perimeter of a square is a line because each side length is multiplied by 4. The graph representing area of a square is nonlinear because each side length is squared. Squared values do not increase at a constant rate.
28. To find the volume of a cube, multiply its base, its height, and its width.
What is the volume of the cube expressed as a power?
A. 6 2 in
3
SOLUTION: A cube has the same measures for all sides, so the base, height, and width are all 6 inches. The volume
is 6 • 6 • 6 or 6 3 inches cubed. This corresponds to
choice B.
29. Short Response The volume of an ice cube in
cubic millimeters is represented by the term 11 3 .
What is 11 3 in standard form?
SOLUTION:
30. What is the value of x2 – y 4 if x = –3 and y = –2?
F. –7 G. –2 H. 2 I. 7
SOLUTION:
Replace x with –3 and y with –2 in the expression
x2 – y 4. Then write the powers as products and simplify.
The correct answer is F.
31. The table below shows the number of ants in an ant farm on different days. The number of ants doubles every ten days.
a. How many ants were in the farm on Day 1? b. How many ants will be in the farm on Day 91?
SOLUTION: a. Use the look for a pattern strategy. The number of ants doubles every 10 days, so going backward, the number of ants is halved each time the number of days is decreased by 10. Extend the pattern to find the number of ants on Day 1.
So, there were 10 ants in the farm on Day 1. b. Extend the pattern to find the number of ants in the farm on Day 91. The number of ants doubles every 10 days.
So, there will be 5,120 ants in the farm on Day 91.
Day 61 51 41 31 21 11 1 Number of ants
640 320 160 80 40 20 10
Day 51 61 71 81 91 Number of ants
320 640 1,280 2,560 5,120
32. Nieves and her three friends are playing a video game. The table shows their scores at the end of the first round.
a. What is the difference between the highest and lowest scores? b. By how many points is Nieves losing to Polly?
SOLUTION: a. The highest score is 230 points. The lowest score is –189. Subtract: 230 – (–189) = 230 + 189 or 419 The difference between the highest and lowest scores is 419 points. b. Nieves' score is –189. Polly's score is –142. Subtract: –142 – (–189) = –142 + 189 or 47 Polly is ahead of Nieves by 47 points.
Add. 33. –12 + (–19)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –12 + (–19) = –31
34. –8 + (–11)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –8 + (–11) = –19
35. –5 + 6
SOLUTION: Since the signs are different, subtract the absolute values of the numbers. Since the largest absolute value is 6, the sign of the answer is the same as the sign of the 6. –5 + 6 = 1
eSolutions Manual - Powered by Cognero Page 3
1-2 Powers and Exponents
Write the expression using exponents. 1. (–5)(–5)(–5)(–5)
SOLUTION: The base –5 is a factor 4 times, so the exponent is 4.
(–5)(–5)(–5)(–5) = (–5) 4
2. 3 • 3 • 5 • q • q • q
SOLUTION: The base 3 is a factor 2 times, so its exponent is 2. The base 5 is a factor 1 time, so its exponent is 1. Because a number without an exponent is the same as a number with an exponent of 1, write 5 without an exponent. The base q is a factor 3 times, so its exponent is 3.
3 • 3 • 5 • q • q • q = 3 2 • 5 • q3
3. m • m • m • m • m
SOLUTION: The base m is a factor 5 times, so its exponent is 5.
m • m • m • m • m = m 5
Evaluate the expression.
4. (–9) 4
(–9) 4 = (–9) • (–9) • (–9) • (–9) or 6,561
5.
or
6.
7. In the United States, nearly 8 • 10 9 text messages
are sent every month. About how many text messages is this?
SOLUTION: Write the power as a product and multiply.
There are about 8 billion text messages sent in the United States each month.
8. Interstate 70 stretches almost 2 3 • 5
2 • 11 miles
across the United States. About how many miles long is Interstate 70?
SOLUTION: Write the power as a product and multiply.
Interstate 70 stretches for almost 2,200 miles.
Evaluate the expression.
9. g5 – h 3 if g = 2 and h = 7
SOLUTION: Replace g with 2 and h with 7 in the expression
. Then write the powers as products and simplify.
10. c 2 + d
3 if c = 8 and d = –3
SOLUTION: Replace c with 8 and d with –3 in the expression
. Then write the powers as products and simplify.
11. a 2 • b
SOLUTION:
Replace a with and b with 2 in the expression
. Then write the powers as products and simplify.
12. (r – s) 3 + r
2 if r = –3 and s = –4
SOLUTION: Replace r with –3 and s with –4 in the expression
. Then write the powers as products and simplify.
13. Model with Mathematics Refer to the graphic novel frame below for Exercises a–c.
The metric system is based on powers of 10. For example, one kilometer is equal to 1,000 meters or
10 3 meters. Write each measurement in meters as a
power of 10. a. megameter (1,000,000 meters) b. gigameter (1,000,000,000 meters) c. petameter (1,000,000,000,000,000 meters)
SOLUTION: a. megameter: 1,000,000 meters has 6 zeros or
powers of 10. So, a megameter is 10 6 .
b. gigameter: 1,000,000,000 meters has 9 zeros or
powers of 10. So, a gigameter is 10 9 .
c. petameter: 1,000,000,000,000,000 meters has 15
.
14. Identify Structure Write an expression with an exponent that has a value between 0 and 1.
SOLUTION:
3 = 27, 3
2 = 9, 3
1 = 3.
.
1 = 3
To go from 81 to 27, divide 81 by 3. This is the same for the rest of the values: 27 ÷ 3 = 9, and 9 ÷ 3 = 3.
To find the value of 2 -1
, follow the same pattern.
2 3 = 8, 2
0 = 2 ÷ 2 or
2 3 • 3
4
A. 3 • 3 • 4 • 4 • 4 B. 2 • 2 • 2 • 3 • 3 • 3 C. 2 • 2 • 2 • 3 • 3 • 3 • 3 D. 6 • 12
SOLUTION:
2 3 • 34
= 2 • 2 • 2 • 3 • 3 • 3 • 3. This corresponds to choice C.
17. Write 3 · p p p · 3 · 3 using exponents.
18. Evaluate x3 + y 4 if x = −3 and y = 4.
SOLUTION: Replace x with –3 and y with 4 in the expression
. Then write the powers as products and simplify.
Write the expression using exponents.
19.
SOLUTION:
The base is a factor 3 times, so its exponent is
3.
20. s • (–7) • s • (–7) • (–7)
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base –7 is a factor 3 times, so its exponent is 3. The base s is a factor 2 times, so its exponent is 2.
21. 4 • b • b • 4 • b • b
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base 4 is a factor 2 times, so its exponent is 2. The base b is a factor 4 times, so its exponent is 4.
Evaluate the expression.
SOLUTION:
Replace k with 3 and m with in the expression
k 4 • m. Then write the powers as products and
simplify.
3 , if c = –1 and d = 2
SOLUTION: Replace c with –1 and d with 2 in the expression
. Then write the powers as
products and simplify.
statement.
2
SOLUTION: Simplify each expression.
28 is larger than 25, so > will make the statement true.
(6 – 2) 2 + 3 • 4 > 5
2
3 3
SOLUTION: Simplify each expression.
81 is equal to 81, so = will make the statement true.
5 + 7 2 + 3
true.
=
27. Multiple Representations A square has a side length of s inches. a. Tables Copy and complete the table showing the side length, perimeter, and area of the square on a separate piece of paper.
b. Graphs On a separate piece of grid paper, graph the ordered pairs (side length, perimeter) and (side length, area) on the same coordinate plane. Then connect the points for each set. c. Words On a separate piece of paper, compare and contrast the graphs of the perimeter and area of the square. Which graph is a line?
SOLUTION: a. The perimeter of a square is found using the expression 4s. The area of a square is found using
the expression s 2 . Complete the table.
b. Graph the values from the table on the same coordinate plane. Side length is represented on the x- axis and both perimeter and area are represented on the y-axis.
c. Sample answer: The graph representing perimeter of a square is a line because each side length is multiplied by 4. The graph representing area of a square is nonlinear because each side length is squared. Squared values do not increase at a constant rate.
28. To find the volume of a cube, multiply its base, its height, and its width.
What is the volume of the cube expressed as a power?
A. 6 2 in
3
SOLUTION: A cube has the same measures for all sides, so the base, height, and width are all 6 inches. The volume
is 6 • 6 • 6 or 6 3 inches cubed. This corresponds to
choice B.
29. Short Response The volume of an ice cube in
cubic millimeters is represented by the term 11 3 .
What is 11 3 in standard form?
SOLUTION:
30. What is the value of x2 – y 4 if x = –3 and y = –2?
F. –7 G. –2 H. 2 I. 7
SOLUTION:
Replace x with –3 and y with –2 in the expression
x2 – y 4. Then write the powers as products and simplify.
The correct answer is F.
31. The table below shows the number of ants in an ant farm on different days. The number of ants doubles every ten days.
a. How many ants were in the farm on Day 1? b. How many ants will be in the farm on Day 91?
SOLUTION: a. Use the look for a pattern strategy. The number of ants doubles every 10 days, so going backward, the number of ants is halved each time the number of days is decreased by 10. Extend the pattern to find the number of ants on Day 1.
So, there were 10 ants in the farm on Day 1. b. Extend the pattern to find the number of ants in the farm on Day 91. The number of ants doubles every 10 days.
So, there will be 5,120 ants in the farm on Day 91.
Day 61 51 41 31 21 11 1 Number of ants
640 320 160 80 40 20 10
Day 51 61 71 81 91 Number of ants
320 640 1,280 2,560 5,120
32. Nieves and her three friends are playing a video game. The table shows their scores at the end of the first round.
a. What is the difference between the highest and lowest scores? b. By how many points is Nieves losing to Polly?
SOLUTION: a. The highest score is 230 points. The lowest score is –189. Subtract: 230 – (–189) = 230 + 189 or 419 The difference between the highest and lowest scores is 419 points. b. Nieves' score is –189. Polly's score is –142. Subtract: –142 – (–189) = –142 + 189 or 47 Polly is ahead of Nieves by 47 points.
Add. 33. –12 + (–19)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –12 + (–19) = –31
34. –8 + (–11)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –8 + (–11) = –19
35. –5 + 6
SOLUTION: Since the signs are different, subtract the absolute values of the numbers. Since the largest absolute value is 6, the sign of the answer is the same as the sign of the 6. –5 + 6 = 1
Write the expression using exponents. 1. (–5)(–5)(–5)(–5)
SOLUTION: The base –5 is a factor 4 times, so the exponent is 4.
(–5)(–5)(–5)(–5) = (–5) 4
2. 3 • 3 • 5 • q • q • q
SOLUTION: The base 3 is a factor 2 times, so its exponent is 2. The base 5 is a factor 1 time, so its exponent is 1. Because a number without an exponent is the same as a number with an exponent of 1, write 5 without an exponent. The base q is a factor 3 times, so its exponent is 3.
3 • 3 • 5 • q • q • q = 3 2 • 5 • q3
3. m • m • m • m • m
SOLUTION: The base m is a factor 5 times, so its exponent is 5.
m • m • m • m • m = m 5
Evaluate the expression.
4. (–9) 4
(–9) 4 = (–9) • (–9) • (–9) • (–9) or 6,561
5.
or
6.
7. In the United States, nearly 8 • 10 9 text messages
are sent every month. About how many text messages is this?
SOLUTION: Write the power as a product and multiply.
There are about 8 billion text messages sent in the United States each month.
8. Interstate 70 stretches almost 2 3 • 5
2 • 11 miles
across the United States. About how many miles long is Interstate 70?
SOLUTION: Write the power as a product and multiply.
Interstate 70 stretches for almost 2,200 miles.
Evaluate the expression.
9. g5 – h 3 if g = 2 and h = 7
SOLUTION: Replace g with 2 and h with 7 in the expression
. Then write the powers as products and simplify.
10. c 2 + d
3 if c = 8 and d = –3
SOLUTION: Replace c with 8 and d with –3 in the expression
. Then write the powers as products and simplify.
11. a 2 • b
SOLUTION:
Replace a with and b with 2 in the expression
. Then write the powers as products and simplify.
12. (r – s) 3 + r
2 if r = –3 and s = –4
SOLUTION: Replace r with –3 and s with –4 in the expression
. Then write the powers as products and simplify.
13. Model with Mathematics Refer to the graphic novel frame below for Exercises a–c.
The metric system is based on powers of 10. For example, one kilometer is equal to 1,000 meters or
10 3 meters. Write each measurement in meters as a
power of 10. a. megameter (1,000,000 meters) b. gigameter (1,000,000,000 meters) c. petameter (1,000,000,000,000,000 meters)
SOLUTION: a. megameter: 1,000,000 meters has 6 zeros or
powers of 10. So, a megameter is 10 6 .
b. gigameter: 1,000,000,000 meters has 9 zeros or
powers of 10. So, a gigameter is 10 9 .
c. petameter: 1,000,000,000,000,000 meters has 15
.
14. Identify Structure Write an expression with an exponent that has a value between 0 and 1.
SOLUTION:
3 = 27, 3
2 = 9, 3
1 = 3.
.
1 = 3
To go from 81 to 27, divide 81 by 3. This is the same for the rest of the values: 27 ÷ 3 = 9, and 9 ÷ 3 = 3.
To find the value of 2 -1
, follow the same pattern.
2 3 = 8, 2
0 = 2 ÷ 2 or
2 3 • 3
4
A. 3 • 3 • 4 • 4 • 4 B. 2 • 2 • 2 • 3 • 3 • 3 C. 2 • 2 • 2 • 3 • 3 • 3 • 3 D. 6 • 12
SOLUTION:
2 3 • 34
= 2 • 2 • 2 • 3 • 3 • 3 • 3. This corresponds to choice C.
17. Write 3 · p p p · 3 · 3 using exponents.
18. Evaluate x3 + y 4 if x = −3 and y = 4.
SOLUTION: Replace x with –3 and y with 4 in the expression
. Then write the powers as products and simplify.
Write the expression using exponents.
19.
SOLUTION:
The base is a factor 3 times, so its exponent is
3.
20. s • (–7) • s • (–7) • (–7)
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base –7 is a factor 3 times, so its exponent is 3. The base s is a factor 2 times, so its exponent is 2.
21. 4 • b • b • 4 • b • b
SOLUTION: Use the Commutative and Associative Properties to group the factors. The base 4 is a factor 2 times, so its exponent is 2. The base b is a factor 4 times, so its exponent is 4.
Evaluate the expression.
SOLUTION:
Replace k with 3 and m with in the expression
k 4 • m. Then write the powers as products and
simplify.
3 , if c = –1 and d = 2
SOLUTION: Replace c with –1 and d with 2 in the expression
. Then write the powers as
products and simplify.
statement.
2
SOLUTION: Simplify each expression.
28 is larger than 25, so > will make the statement true.
(6 – 2) 2 + 3 • 4 > 5
2
3 3
SOLUTION: Simplify each expression.
81 is equal to 81, so = will make the statement true.
5 + 7 2 + 3
true.
=
27. Multiple Representations A square has a side length of s inches. a. Tables Copy and complete the table showing the side length, perimeter, and area of the square on a separate piece of paper.
b. Graphs On a separate piece of grid paper, graph the ordered pairs (side length, perimeter) and (side length, area) on the same coordinate plane. Then connect the points for each set. c. Words On a separate piece of paper, compare and contrast the graphs of the perimeter and area of the square. Which graph is a line?
SOLUTION: a. The perimeter of a square is found using the expression 4s. The area of a square is found using
the expression s 2 . Complete the table.
b. Graph the values from the table on the same coordinate plane. Side length is represented on the x- axis and both perimeter and area are represented on the y-axis.
c. Sample answer: The graph representing perimeter of a square is a line because each side length is multiplied by 4. The graph representing area of a square is nonlinear because each side length is squared. Squared values do not increase at a constant rate.
28. To find the volume of a cube, multiply its base, its height, and its width.
What is the volume of the cube expressed as a power?
A. 6 2 in
3
SOLUTION: A cube has the same measures for all sides, so the base, height, and width are all 6 inches. The volume
is 6 • 6 • 6 or 6 3 inches cubed. This corresponds to
choice B.
29. Short Response The volume of an ice cube in
cubic millimeters is represented by the term 11 3 .
What is 11 3 in standard form?
SOLUTION:
30. What is the value of x2 – y 4 if x = –3 and y = –2?
F. –7 G. –2 H. 2 I. 7
SOLUTION:
Replace x with –3 and y with –2 in the expression
x2 – y 4. Then write the powers as products and simplify.
The correct answer is F.
31. The table below shows the number of ants in an ant farm on different days. The number of ants doubles every ten days.
a. How many ants were in the farm on Day 1? b. How many ants will be in the farm on Day 91?
SOLUTION: a. Use the look for a pattern strategy. The number of ants doubles every 10 days, so going backward, the number of ants is halved each time the number of days is decreased by 10. Extend the pattern to find the number of ants on Day 1.
So, there were 10 ants in the farm on Day 1. b. Extend the pattern to find the number of ants in the farm on Day 91. The number of ants doubles every 10 days.
So, there will be 5,120 ants in the farm on Day 91.
Day 61 51 41 31 21 11 1 Number of ants
640 320 160 80 40 20 10
Day 51 61 71 81 91 Number of ants
320 640 1,280 2,560 5,120
32. Nieves and her three friends are playing a video game. The table shows their scores at the end of the first round.
a. What is the difference between the highest and lowest scores? b. By how many points is Nieves losing to Polly?
SOLUTION: a. The highest score is 230 points. The lowest score is –189. Subtract: 230 – (–189) = 230 + 189 or 419 The difference between the highest and lowest scores is 419 points. b. Nieves' score is –189. Polly's score is –142. Subtract: –142 – (–189) = –142 + 189 or 47 Polly is ahead of Nieves by 47 points.
Add. 33. –12 + (–19)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –12 + (–19) = –31
34. –8 + (–11)
SOLUTION: Since the signs are the same, add the numbers. The sign of the answer is the same as the sign of addends. –8 + (–11) = –19
35. –5 + 6
SOLUTION: Since the signs