VOCABULARY
Order of magnitude
Copyright © Holt McDougal. All rights reserved. Lesson 8.1 • Algebra 1 Notetaking Guide 193
Your Notes
PRODUCT OF POWERS PROPERTY
Let a be a real number, and let m and n be positive integers.
Words: To multiply powers having the same base, .
Algebra: am p an 5 a
Example: 56 p 53 5 5 5 5
Simplify the expression.
a. 22 p 23 5 2
5 2
b. w9 p w2 p w7 5 w
5 w
c. 44 p 4 5 44 p 4
5 4
5 4
d. (26)(26)6 5 (26) p (26)6
5 (26)
5 (26)
Example 1 Use the product of powers property
When simplifying powers with numerical bases only, write your answers using exponents.
Apply Exponent PropertiesInvolving ProductsGoal p Use properties of exponents involving products.
8.1
VOCABULARY
Order of magnitude The order of magnitude of a quantity is the power of 10 nearest the quantity.
Copyright © Holt McDougal. All rights reserved. Lesson 8.1 • Algebra 1 Notetaking Guide 193
Your Notes
PRODUCT OF POWERS PROPERTY
Let a be a real number, and let m and n be positive integers.
Words: To multiply powers having the same base, add the exponents .
Algebra: am p an 5 a m 1 n
Example: 56 p 53 5 5 6 1 3 5 5 9
Simplify the expression.
a. 22 p 23 5 2 2 1 3
5 2 5
b. w9 p w2 p w7 5 w 9 1 2 1 7
5 w 18
c. 44 p 4 5 44 p 4 1
5 4 4 1 1
5 4 5
d. (26)(26)6 5 (26) 1 p (26)6
5 (26) 1 1 6
5 (26) 7
Example 1 Use the product of powers property
When simplifying powers with numerical bases only, write your answers using exponents.
Apply Exponent PropertiesInvolving ProductsGoal p Use properties of exponents involving products.
8.1
Your NotesPOWER OF A POWER PROPERTY
Let a be a real number, and let m and n be positive integers.
Words: To find a power of a power, .
Algebra: (am)n 5 a
Example: (34)2 5 3 5 3
Simplify the expression.
a. (52)3 5 5 5 5
b. (n7)2 5 n 5 n
c. [(23)5]3 5 (23)
5 (23)
d. [(z 2 4)2]5 5 (z 2 4)
5 (z 2 4)
Example 2 Use the power of a power property
POWER OF A PRODUCT PROPERTY
Let a and b be real numbers, and let m be a positive integer.
Words: To find a power of a product, find the .
Algebra: (ab)m 5
Example: (23 p 17)5 5
Simplify the expression.
a. (4 p 16)7 5
b. (23rs)2 5 ( )2 5 ( )2 p 2 p 2
5
c. 2(3rs)2 5 2( )2 5 2( 2 p 2 p 2) 5
Example 3 Use the power of a product property
When simplifying powers with numerical and variable bases, evaluate the numerical power.
194 Lesson 8.1 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your NotesPOWER OF A POWER PROPERTY
Let a be a real number, and let m and n be positive integers.
Words: To find a power of a power, multiply exponents .
Algebra: (am)n 5 a mn
Example: (34)2 5 3 4 p 2 5 3 8
Simplify the expression.
a. (52)3 5 5 2 p 3 5 5 6
b. (n7)2 5 n 7 p 2 5 n 14
c. [(23)5]3 5 (23) 5 p 3
5 (23) 15
d. [(z 2 4)2]5 5 (z 2 4) 2 p 5
5 (z 2 4) 10
Example 2 Use the power of a power property
POWER OF A PRODUCT PROPERTY
Let a and b be real numbers, and let m be a positive integer.
Words: To find a power of a product, find the power of each factor and multiply .
Algebra: (ab)m 5 ambm
Example: (23 p 17)5 5 235 p 175
Simplify the expression.
a. (4 p 16)7 5 47 p 167
b. (23rs)2 5 ( 23 p r p s )2 5 ( 23 )2 p r 2 p s 2
5 9r2s2
c. 2(3rs)2 5 2( 3 p r p s )2 5 2( 3 2 p r 2 p s 2) 5 29r2s2
Example 3 Use the power of a product property
When simplifying powers with numerical and variable bases, evaluate the numerical power.
194 Lesson 8.1 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Copyright © Holt McDougal. All rights reserved. Lesson 8.1 • Algebra 1 Notetaking Guide 195
Your Notes
1. (27)8(27)5 2. k3 p k p k2 3. (p3)4
4. [(q 1 8)2]6 5. (8cd)2 6. 2(5z)3
Checkpoint Simplify the expression.
Simplify x2 p (3x3y)3.
Solution
x2 p (3x3y)3 5 property
5 property
5 property
Example 4 Use all three properties
7. (2x5)4 8. (3y3)4 p y5
Checkpoint Simplify the expression.
Homework
Copyright © Holt McDougal. All rights reserved. Lesson 8.1 • Algebra 1 Notetaking Guide 195
Your Notes
1. (27)8(27)5 2. k3 p k p k2 3. (p3)4
(27)13 k6 p12
4. [(q 1 8)2]6 5. (8cd)2 6. 2(5z)3
(q 1 8)12 64c2d2 2125z3
Checkpoint Simplify the expression.
Simplify x2 p (3x3y)3.
Solution
x2 p (3x3y)3 5 x2 p 33 p (x3)3 p y3 Power of a product property
5 x2 p 27 p x9 p y3 Power of a power property
5 27x11y3 Product of powers property
Example 4 Use all three properties
7. (2x5)4 8. (3y3)4 p y5
16x20 81y17
Checkpoint Simplify the expression.
Homework
8.2 Apply Exponent PropertiesInvolving QuotientsGoal p Use properties of exponents involving quotients.
QUOTIENT OF POWERS PROPERTY
Let a be a nonzero real number, and let m and n be positive integers such that m > n.
Words: To divide powers having the same base, the exponents.
Algebra: am }
an 5 a , a Þ 0
Example: 47 }
42 5 4 5 4
Simplify the expression.
a. 612 }
65 5 6 5 6
b. (22)7
} (22)4
5 (22) 5 (22)
c. 42 p 48
} 44 5 4}
44
5 4
5
d. 1 } y9 p y12 5
y12 }
y9
5 y
5
Example 1 Use the quotient of powers property
When simplifying powers with numerical bases only, write your answers using exponents.
Your Notes
196 Lesson 8.2 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
8.2 Apply Exponent PropertiesInvolving QuotientsGoal p Use properties of exponents involving quotients.
QUOTIENT OF POWERS PROPERTY
Let a be a nonzero real number, and let m and n be positive integers such that m > n.
Words: To divide powers having the same base, subtract the exponents.
Algebra: am }
an 5 a m 2 n , a Þ 0
Example: 47 }
42 5 4 7 2 2 5 4 5
Simplify the expression.
a. 612 }
65 5 6 12 2 5 5 6 7
b. (22)7
} (22)4
5 (22) 7 2 4 5 (22) 3
c. 42 p 48
} 44 5 4
10 }
44
5 4 10 2 4
5 46
d. 1 } y9 p y12 5
y12 }
y9
5 y 12 2 9
5 y3
Example 1 Use the quotient of powers property
When simplifying powers with numerical bases only, write your answers using exponents.
Your Notes
196 Lesson 8.2 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your NotesPOWER OF A QUOTIENT PROPERTY
Let a and b be real numbers with b Þ 0, and let m be a positive integer.
Words: To find a power of a quotient, find the power of the and the power of the and divide.
Algebra: 1 a } b 2 m 5 , b Þ 0 Example: 1 4 } 7 2 3 5 ____ ____
Example 2 Use the power of a quotient property
Simplify the expression.
a. 1 r } s 2 5 5
____
b. 1 2 4 } w 2 3 5 1 2 3 5 5 5
____ ______ ______ ______
When simplifying powers with numerical and variable bases, evaluate the numerical power.
1. (28)8
} (28)5
2. 35 p 34
} 33
3. 1 2 r } 3 2 2 4. 1 5 } t 2
4
Checkpoint Simplify the expression.
Copyright © Holt McDougal. All rights reserved. Lesson 8.2 • Algebra 1 Notetaking Guide 197
Your NotesPOWER OF A QUOTIENT PROPERTY
Let a and b be real numbers with b Þ 0, and let m be a positive integer.
Words: To find a power of a quotient, find the power of the numerator and the power of the denominator and divide.
Algebra: 1 a } b 2 m 5 am }
bm , b Þ 0 Example: 1 4 } 7 2 3 5 43 }
73 ____ ____
Example 2 Use the power of a quotient property
Simplify the expression.
a. 1 r } s 2 5 5 r
5 }
s5
____
b. 1 2 4 } w 2 3 5 1 24 } w 2 3 5 (24)3 }
w3 5 264 }
w3 5 2 64 } w3
____ ______ ______ ______
When simplifying powers with numerical and variable bases, evaluate the numerical power.
1. (28)8
} (28)5
2. 35 p 34
} 33
(28)3 36
3. 1 2 r } 3 2 2 4. 1 5 } t 2
4
r2 } 9 625 }
t4
Checkpoint Simplify the expression.
Copyright © Holt McDougal. All rights reserved. Lesson 8.2 • Algebra 1 Notetaking Guide 197
Your Notes
Homework
Example 3 Use properties of exponents
Simplify 1 2y7 }
y5 2 3.
Solution
1 2y7 }
y5 2 3 5 property
5 property
5 property
5 property
5. 1 7y3z } y 2 2 6. 2s4
} t p 1 2t } s 2 3
7. 1 6m3n2 } 3mn 2 3 8. 4a }
b2 p 1 2a2b3 } a 2 4
Checkpoint Simplify the expression.
198 Lesson 8.2 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
Homework
Example 3 Use properties of exponents
Simplify 1 2y7 }
y5 2 3.
Solution
1 2y7 }
y5 2 3 5 (2y7)3
} (y5)3
Power of a quotient property
5 23 p (y7)3
} (y5)3
Power of a product property
5 8y21
} y15
Power of a power property
5 8y6 Quotient of powers property
5. 1 7y3z } y 2 2 6. 2s4
} t p 1 2t } s 2 3
49y5z2 16st2
7. 1 6m3n2 } 3mn 2 3 8. 4a }
b2 p 1 2a2b3 } a 2 4
8m6n3 64a5b10
Checkpoint Simplify the expression.
198 Lesson 8.2 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
8.3 Define and Use Zero andNegative ExponentsGoal p Use zero and negative exponents.
DEFINITION OF ZERO AND NEGATIVE EXPONENTS
Words Algebra Example
a to the zero power a0 5 , a Þ 0 50 5 is 1.
a2n is the reciprocal a2n 5 , a Þ 0 221 5 of an. ___ ____
an is the reciprocal an 5 , a Þ 0 2 5 of a2n. ____ ____
Example 1 Use definition of zero and negative exponents
Evaluate the expression.
a. 223 5 Definition of ____
5 Evaluate exponent. ___
b. (210)0 5 Definition of
c. 1 1 } 4 2 23
5 Definition of
____
5 Evaluate exponent.
___
5 Simplify.
d. 027 5 a2n is defined only for a number a.
Copyright © Holt McDougal. All rights reserved. Lesson 8.3 • Algebra 1 Notetaking Guide 199
Your Notes
8.3 Define and Use Zero andNegative ExponentsGoal p Use zero and negative exponents.
DEFINITION OF ZERO AND NEGATIVE EXPONENTS
Words Algebra Example
a to the zero power a0 5 1 , a Þ 0 50 5 1 is 1.
a2n is the reciprocal a2n 5 1 } an , a Þ 0 221 5 1 }
2
of an. ___ ____
an is the reciprocal an 5 1 } a2n , a Þ 0 2 5 1 }
221 of a2n. ____ ____
Example 1 Use definition of zero and negative exponents
Evaluate the expression.
a. 223 5 1 } 23
Definition of negative ____ exponents
5 1 } 8 Evaluate exponent.
___
b. (210)0 5 1 Definition of zero exponent
c. 1 1 } 4 2 23
5 1 } 1 1 } 4 2 3
Definition of negative
____ exponents
5 1 } 1 } 64
Evaluate exponent.
___
5 64 Simplify.
d. 027 5 undefined a2n is defined only for a nonzero number a.
Copyright © Holt McDougal. All rights reserved. Lesson 8.3 • Algebra 1 Notetaking Guide 199
Your Notes
Your NotesPROPERTIES OF EXPONENTS
Let a and b be real numbers, and let m and n be integers.
am p an 5 a property
(am)n 5 a property
(ab)m 5 property
am }
an 5 a , a Þ 0 property
1 a } b 2 m 5 , b Þ 0 property ____
Evaluate the expression.
a. (25)4 p (25)24 5 Product of powersproperty
5 exponents.
5 Definition of
b. (522)22 5 property
5 exponents.
5 Evaluate power.
c. 1 } 422 5 Definition of
5 Evaluate power.
d. 32 }
321 5 property
5 exponents.
5 Evaluate power.
Example 2 Evaluate exponential expressions
200 Lesson 8.3 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your NotesPROPERTIES OF EXPONENTS
Let a and b be real numbers, and let m and n be integers.
am p an 5 a m 1 n Product of powers property
(am)n 5 a mn Power of a power property
(ab)m 5 ambm Power of a product property
am }
an 5 a m 2 n , a Þ 0 Quotient of powers property
1 a } b 2 m 5 am }
bm , b Þ 0 Power of a quotient property ____
Evaluate the expression.
a. (25)4 p (25)24 5 (25)4 1 (24) Product of powersproperty
5 50 Add exponents.
5 1 Definition of zero exponent
b. (522)22 5 522 p (22) Power of a power property
5 54 Multiply exponents.
5 625 Evaluate power.
c. 1 } 422 5 42 Definition of
negative exponents
5 16 Evaluate power.
d. 32 }
321 5 32 2 (21) Quotient of powers property
5 33 Subtract exponents.
5 27 Evaluate power.
Example 2 Evaluate exponential expressions
200 Lesson 8.3 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
Homework
1. 1 1 } 8 2 21 2. 1 } 322
3. 621 } 6 4. (521)2
Checkpoint Evaluate the expression.
Example 3 Use properties of exponents
Simplify the expression 2w23x } (2wx)2
. Write your answer using
only positive exponents.
Solution
2w23x } (2wx)2
5 Definition of negative exponents ___________
5 property ___________
5 property ___________
5 property ___________
5. 6fg24
} 2f2g
6. (3yz2)22
Checkpoint Simplify the expression.
Copyright © Holt McDougal. All rights reserved. Lesson 8.3 • Algebra 1 Notetaking Guide 201
Your Notes
Homework
1. 1 1 } 8 2 21 2. 1 } 322
8 9
3. 621 } 6 4. (521)2
1 } 36 1 } 25
Checkpoint Evaluate the expression.
Example 3 Use properties of exponents
Simplify the expression 2w23x } (2wx)2
. Write your answer using
only positive exponents.
Solution
2w23x } (2wx)2
5 2x } w3(2wx)2
Definition of negative exponents ___________
5 2x } w3(4w2x2)
Power of a product property ___________
5 2x } 4w5x2
Product of powers property ___________
5 1 } 2w5x
Quotient of powers property ___________
5. 6fg24
} 2f2g
6. (3yz2)22
3 } fg5 1 }
9y2z4
Checkpoint Simplify the expression.
Copyright © Holt McDougal. All rights reserved. Lesson 8.3 • Algebra 1 Notetaking Guide 201
Define and Use Fractional Exponents
Goal p Use fractional exponents.
a. 25 1 } 2
5 b. 362 1 } 2
5
5 5
5
c. 4 5 } 2
5 d. 492 3 } 2
5
5 5
5 5
5 5
5 5
5
Example 1 Evaluate expressions involving square roots
1. 16 3 } 2 2. 64
2 1 } 2
3. 1442 3 } 2
4. 25 5 } 2
Checkpoint Evaluate the expression.
202 8.3 Focus On Operations • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
Focus On OperationsUse after Lesson 8.3
Define and Use Fractional Exponents
Goal p Use fractional exponents.
a. 25 1 } 2
5 Ï}
25 b. 362 1 } 2
5 1 }
36 1 } 2
5 5 5 1 }
Ï}
36
5 1 } 6
c. 4 5 } 2
5 4 1 1 } 2 2 p 5
d. 492 3 } 2
5 49 1 1 } 2 2 p 1 23 2
5 1 4 1 } 2 2 5 5 1 49
1 } 2 2 23
5 1 Ï}
4 2 5 5 1 Ï}
49 2 23
5 25 5 723
5 32 5 1 } 73
5 1 } 343
Example 1 Evaluate expressions involving square roots
1. 16 3 } 2 64 2. 64
2 1 } 2 1 }
8
3. 1442 3 } 2
1 } 1728
4. 25 5 } 2 3125
Checkpoint Evaluate the expression.
202 8.3 Focus On Operations • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
Focus On OperationsUse after Lesson 8.3
Homework
Your Notes
5. 64 2 } 3 6. 125
2 4 } 3
7. 1 1 } 4 2 2 3 } 2
1 1 } 4 2
5 } 2 8. 1 27 2
5 } 3 1 27 2
2 4 } 3
Checkpoint Evaluate the expression.
Example 3 Use properties of exponents
a. 152 1 } 2 p 15 5 } 2
5 b. 4 4 } 3 p 4 }
4 1 } 3 5
5 5
5 5
5 5
5
Copyright © Holt McDougal. All rights reserved. 8.3 Focus On Operations • Algebra 1 Notetaking Guide 203
a. 64 1 } 3 5 b. 216
2 1 } 3 5
5 5
5 5
c. 8 4 } 3 5 d. 27
2 2 } 3 5
5 5
5 5
5 5
5 5
5
Example 2 Evaluate expressions involving cube roots
Homework
Your Notes
5. 64 2 } 3 16 6. 125
2 4 } 3 1 }
625
7. 1 1 } 4 2 2 3 } 2
1 1 } 4 2
5 } 2 1 }
4 8. 1 27 2
5 } 3 1 27 2
2 4 } 3 3
Checkpoint Evaluate the expression.
Example 3 Use properties of exponents
a. 152 1 } 2 p 15 5 } 2
515 1 2
1 } 2 2 1
1
5 } 2 2
b. 4 4 } 3 p 4 }
4 1 } 3 5 4
1 4 } 3 2 11
}
4 1 } 3
5 15 4 } 2 5 4
1 7 } 3 2 }
4 1 } 3
5 152 5 4 1 7 } 3 2 2 1 1 }
3 2
5 225 5 42
5 16
Copyright © Holt McDougal. All rights reserved. 8.3 Focus On Operations • Algebra 1 Notetaking Guide 203
a. 64 1 } 3 5
3 Ï}
64 b. 2162 1 } 3
5 1 }
216 1 } 3
5 3 Ï}
43 5 1 }
3 Ï}
216
5 4 5 1 } 6
c. 8 4 } 3 5 8 1
1 } 2 2 p 4 d. 27
2 2 } 3 5 27 1 1 } 3 2 p 1 22 2
5 1 8 1 } 3 2 4 5 1 27
1 } 3 2 22
5 1 3 Ï}
8 2 4 5 1 3 Ï}
27 2 22
5 24 5 322
516 5 1 }
32
5 1 } 9
Example 2 Evaluate expressions involving cube roots
8.4 Use Scientific NotationGoal p Read and write numbers in scientific notation.
VOCABULARY
Scientific notation Your Notes
SCIENTIFIC NOTATION
A number is written in scientific notation when it is of the form where 1 ≤ c < 10 and n is an integer.
Number Standard form Scientific notation
Sixteen million
Two hundredths
a. 7,820,000 5 3 10 Move decimal point places to the . Exponent is .
b. 0.00401 5 3 10 Move decimal point places to the . Exponent is .
Example 1 Write numbers in scientific notation
a. 3.89 3 109 5 Exponent is . Move decimal point
places to the .
b. 9.097 3 1025 5 Exponent is . Move decimal point places to the
.
Example 2 Write numbers in standard form
204 Lesson 8.4 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
8.4 Use Scientific NotationGoal p Read and write numbers in scientific notation.
VOCABULARY
Scientific notation A number is written in scientific notation when it is of the form c 3 10n where 1 ≤ c < 10 and n is an integer.
Your Notes
SCIENTIFIC NOTATION
A number is written in scientific notation when it is of the form c 3 10n where 1 ≤ c < 10 and n is an integer.
Number Standard form Scientific notation
Sixteen million 16,000,000 1.6 3 107
Two hundredths 0.02 2 3 1022
a. 7,820,000 5 7.82 3 10 6 Move decimal point 6 places to the left . Exponent is 6 .
b. 0.00401 5 4.01 3 10 23 Move decimal point 3 places to the right . Exponent is 23 .
Example 1 Write numbers in scientific notation
a. 3.89 3 109 5 3,899,000,000 Exponent is 9 . Move decimal point 9 places to the right .
b. 9.097 3 1025 5 0.00009097 Exponent is 25 . Move decimal point 5 places to the left .
Example 2 Write numbers in standard form
204 Lesson 8.4 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
Copyright © Holt McDougal. All rights reserved. Lesson 8.4 • Algebra 1 Notetaking Guide 205
1. Write the number 0.0899 in scientific notation. Then write the number 6.0001 3 107 in standard form.
Checkpoint Complete the following exercise.
Order 3.2 3 1024, 0.0004, and 2.8 3 1025 from least to greatest.
SolutionStep 1 Write each number in scientific notation, if
necessary.
0.0004 5
Step 2 Order the numbers. First order the numbers with different powers of 10. Then order the numbers with the same power of 10.
Because 1025 1024, you know that is less than both and
. Because 3.2 4, you know that is less than .
So, < < .
Step 3 Write the original numbers in order from least to greatest.
Example 3 Order numbers in scientific notation
2. Order 225,000, 1,740,000, and 1.75 3 105 from least to greatest.
Checkpoint Complete the following exercise.
Your Notes
Copyright © Holt McDougal. All rights reserved. Lesson 8.4 • Algebra 1 Notetaking Guide 205
1. Write the number 0.0899 in scientific notation. Then write the number 6.0001 3 107 in standard form.
8.99 3 1022; 60,001,000
Checkpoint Complete the following exercise.
Order 3.2 3 1024, 0.0004, and 2.8 3 1025 from least to greatest.
SolutionStep 1 Write each number in scientific notation, if
necessary.
0.0004 5 4 3 1024
Step 2 Order the numbers. First order the numbers with different powers of 10. Then order the numbers with the same power of 10.
Because 1025 < 1024, you know that 2.8 3 1025 is less than both 3.2 3 1024 and 4 3 1024 . Because 3.2 < 4, you know that 3.2 3 1024 is less than 4 3 1024 .
So, 2.8 3 1025 < 3.2 3 1024 < 4 3 1024 .
Step 3 Write the original numbers in order from least to greatest.
2.8 3 1025; 3.2 3 1024; 0.0004
Example 3 Order numbers in scientific notation
2. Order 225,000, 1,740,000, and 1.75 3 105 from least to greatest.
1.75 3 105; 225,000; 1,740,000
Checkpoint Complete the following exercise.
Your Notes
Evaluate the expression. Write your answer in scientific notation.
a. (5.6 3 1024)(1.4 3 1025) 5 (5.6 p 1.4) 3 (1024 p 1025) Commutative property
and associative property
5 3 Product of powers property
b. (3.2 3 102)3
5 3 Power of a product property
5 3 Power of a power property
5 ( ) 3 Write in scientific notation.
5 3 ( ) Associative property
5 Product of powers property
c. 3.5 3 1023 }}
1.75 3 1025
5 3.5 } 1.75 3 1023 }
1025 Product rule for fractions
5 3 Quotient of powers property
Example 4 Compute with numbers in scientific notation
Homework
3. (2.01 3 1027)2 4. 4.8 3 1024 }
6 3 1024
Checkpoint
206 Lesson 8.4 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
Evaluate the expression. Write your answer in scientific notation.
a. (5.6 3 1024)(1.4 3 1025) 5 (5.6 p 1.4) 3 (1024 p 1025) Commutative property
and associative property
5 7.84 3 1029 Product of powers property
b. (3.2 3 102)3
5 3.23 3 (102)3 Power of a product property
5 32.768 3 106 Power of a power property
5 ( 3.2768 3 101 ) 3 106 Write 32.768 in scientific notation.
5 3.2768 3 ( 101 3 106 ) Associative property
5 3.2768 3 107 Product of powers property
c. 3.5 3 1023 }}
1.75 3 1025
5 3.5 } 1.75 3 1023 }
1025 Product rule for fractions
5 2 3 102 Quotient of powers property
Example 4 Compute with numbers in scientific notation
Homework
3. (2.01 3 1027)2 4. 4.8 3 1024 }
6 3 1024
4.0401 3 10214 8 3 1021
Checkpoint
206 Lesson 8.4 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
8.5 Write and Graph ExponentialGrowth FunctionsGoal p Write and graph exponential growth models.
VOCABULARY
Exponential function
Exponential growth
Compound interest
Write a rule for the function.
x 22 21 0 1 2
y 2 } 9 2 } 3 2 6 8
SolutionStep 1 Tell whether the function is exponential. Here the
y-values are multiplied by for each increase of 1 in x, so the table represents an exponential function of the form where .
Step 2 Find the value of a by finding the value of y when x 5 0. When x 5 0, y 5 5 5 . The value of y when x 5 0 is , so .
Step 3 Write the function rule. A rule for the function is y 5 .
Example 1 Write a function rule
Copyright © Holt McDougal. All rights reserved. Lesson 8.5 • Algebra 1 Notetaking Guide 207
Your Notes
8.5 Write and Graph ExponentialGrowth FunctionsGoal p Write and graph exponential growth models.
VOCABULARY
Exponential function A function of the form y 5 abx where a Þ 0, b > 0, and b Þ 1
Exponential growth A quantity that increases by the same percent over equal time periods
Compound interest Interest earned on both an initial investment and on previously earned interest
Write a rule for the function.
x 22 21 0 1 2
y 2 } 9 2 } 3 2 6 8
SolutionStep 1 Tell whether the function is exponential. Here the
y-values are multiplied by 3 for each increase of 1 in x, so the table represents an exponential function of the form y 5 abx where b 5 3 .
Step 2 Find the value of a by finding the value of y when x 5 0. When x 5 0, y 5 ab0 5 a p 1 5 a . The value of y when x 5 0 is 2 , so a 5 2.
Step 3 Write the function rule. A rule for the function is y 5 2 p 3x .
Example 1 Write a function rule
Copyright © Holt McDougal. All rights reserved. Lesson 8.5 • Algebra 1 Notetaking Guide 207
Your Notes
Graph the function y 5 3x. Identify its domain and range.
SolutionStep 1 Make a table by choosing a
x
y
1
3
5
7
9
12123 3
few values for x and finding the values of y. The domain is .
x 22 21 0 1 2
y
Step 2 Plot the points.
Step 3 Draw a smooth curve through the points. From either the table or the graph, you can see that the range is .
Example 2 Graph an exponential function
Graph y 5 2 p 3x. Compare the graph with the graph of y 5 3x.
SolutionTo graph each function, make x y 5 3x y 5 2 p 3x
22
21
0
1
2
a table of values, plot the points, and draw a smooth curve through the points.
x
y
2
6
10
14
18
12123 3
Because the y-values for y 5 2 p 3x are the corresponding y-values for y 5 3x, the graph of y 5 2 p 3x is a of the graph of y 5 3x.
Example 3 Compare graphs of exponential functions
208 Lesson 8.5 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
Graph the function y 5 3x. Identify its domain and range.
SolutionStep 1 Make a table by choosing a
x
y
1
3
5
7
9
12123 3
y 5 3x
(2, 9)
(1, 3)
(0, 1)( )1922,
( )1321,
few values for x and finding the values of y. The domain is all real numbers .
x 22 21 0 1 2
y 1 } 9 1 }
3
1 3 9
Step 2 Plot the points.
Step 3 Draw a smooth curve through the points. From either the table or the graph, you can see that the range is all positive real numbers .
Example 2 Graph an exponential function
Graph y 5 2 p 3x. Compare the graph with the graph of y 5 3x.
SolutionTo graph each function, make x y 5 3x y 5 2 p 3x
22 1 } 9 2 }
9
21 1 } 3 2 }
3
0 1 2
1 3 6
2 9 18
a table of values, plot the points, and draw a smooth curve through the points.
x
y
2
6
10
14
18
12123 3
y 5 3xy 5 2 ? 3x
Because the y-values for y 5 2 p 3x are 2 times the corresponding y-values for y 5 3x, the graph of y 5 2 p 3x is a vertical stretch of the graph of y 5 3x.
Example 3 Compare graphs of exponential functions
208 Lesson 8.5 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
1. Write a rule for the function.
x 22 21 0 1 2
y 2 1 } 16 2 1 } 4 21 24 216
2. Graph y 5 4x. Identify its
x
y
2
6
10
14
12123 322
domain and range.
3. Graph y 5 22 p 3x. Compare
x
y
3
9
12123 323
29
215
the graph with the graph of y 5 3x.
Checkpoint Complete the following exercises.
Copyright © Holt McDougal. All rights reserved. Lesson 8.5 • Algebra 1 Notetaking Guide 209
Your Notes
1. Write a rule for the function.
x 22 21 0 1 2
y 2 1 } 16 2 1 } 4 21 24 216
y 5 21 p 4x
2. Graph y 5 4x. Identify its
x
y
2
6
10
14
12123 3
y 5 4x
22
(0, 1)
(1, 4)
(2, 16)
( )11622,
( )1421,
domain and range.
The domain is all real numbers. The range is all positive real numbers.
3. Graph y 5 22 p 3x. Compare
x
y
3
9
12123 3
y 5 3x
y 5 22 ? 3x23
29
215
the graph with the graph of y 5 3x.
The graph of y 5 22 p 3x is a vertical stretch and a reflection in the x-axis of the graph of y 5 3x.
Checkpoint Complete the following exercises.
Copyright © Holt McDougal. All rights reserved. Lesson 8.5 • Algebra 1 Notetaking Guide 209
Your Notes
Homework
EXPONENTIAL GROWTH MODEL
y 5 a(1 1 r)t
a is the . r is the .
1 1 r is the . t is the .
Investment You put $250 in a savings account that earns 4% annual interest compounded yearly. You do not make any deposits or withdrawals. How much will your investment be worth in 10 years?
Solution
The initial amount is , the interest rate is , or , and the time period is .
y 5 a(1 1 r)t Write exponential growth model.
5 (1 1 ) Substitute for a, for r, and for t.
5 250( )10 Simplify.
ø Use a calculator.
You will have in 10 years.
Example 4 Solve a compound interest problem
4. In Example 4, suppose the annual interest rate is 5%. How much will your investment be worth in 10 years?
Checkpoint Complete the following exercise.
210 Lesson 8.5 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
Homework
EXPONENTIAL GROWTH MODEL
y 5 a(1 1 r)t
a is the initial amount . r is the growth rate .
1 1 r is the growth factor . t is the time period .
Investment You put $250 in a savings account that earns 4% annual interest compounded yearly. You do not make any deposits or withdrawals. How much will your investment be worth in 10 years?
Solution
The initial amount is $250 , the interest rate is 4% , or 0.04 , and the time period is 10 years .
y 5 a(1 1 r)t Write exponential growth model.
5 250 (1 1 0.04 ) 10 Substitute 250 for a, 0.04 for r, and 10 for t.
5 250( 1.04 )10 Simplify.
ø 370.06 Use a calculator.
You will have $370.06 in 10 years.
Example 4 Solve a compound interest problem
4. In Example 4, suppose the annual interest rate is 5%. How much will your investment be worth in 10 years?
about $407.22
Checkpoint Complete the following exercise.
210 Lesson 8.5 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
8.6 Write and Graph ExponentialDecay FunctionsGoal p Write and graph exponential decay functions.
VOCABULARY
Exponential decay
Graph the function y 5 1 1 } 3 2 x and identify its domain and
range.
SolutionStep 1 Make a table of values.
The domain is .
x 22 21 0 1 2
y
Step 2 Plot the points.
x
y
1
3
5
7
9
12123 3
Step 3 Draw a smooth curve through the points. From either the table or the graph, you can see that the range is .
Example 1 Graph an exponential function
Copyright © Holt McDougal. All rights reserved. Lesson 8.6 • Algebra 1 Notetaking Guide 211
Your Notes
8.6 Write and Graph ExponentialDecay FunctionsGoal p Write and graph exponential decay functions.
VOCABULARY
Exponential decay A quantity that decreases by the same percent over equal time periods
Graph the function y 5 1 1 } 3 2 x and identify its domain and
range.
SolutionStep 1 Make a table of values.
The domain is all real numbers .
x 22 21 0 1 2
y 9 3 1 1 } 3 1 }
9
Step 2 Plot the points.
x
y
1
3
5
7
9
12123 3
(22, 9)
(21, 3)
(0, 1) ( )192,
( )131,
y 5x( )1
3
Step 3 Draw a smooth curve through the points. From either the table or the graph, you can see that the range is all positive real numbers .
Example 1 Graph an exponential function
Copyright © Holt McDougal. All rights reserved. Lesson 8.6 • Algebra 1 Notetaking Guide 211
Your Notes
Your Notes
Graph y 5 2 p 1 1 } 3 2 x. Compare the graph with the graph
of y 5 1 1 } 3 2 x.
Solution
x y 5 1 1 } 3 2 x y 5 2 p 1 1 } 3 2 x
22
21
0
1
2
x
y
1
3
5
7
9
12123 3
Because the y-values for y 5 2 p 1 1 } 3 2 x are
the corresponding y-values for y 5 1 1 } 3 2 x, the graph of
y 5 2 p 1 1 } 3 2 x is a of the graph of
y 5 1 1 } 3 2 x.
Example 2 Compare graphs of exponential functions
1. Graph y 5 22 p 1 1 } 3 2 x. Compare the graph with the
graph of 1 1 } 3 2 x.
x
y
3
9
12123 323
29
215
Checkpoint Complete the following exercise.
212 Lesson 8.6 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
Graph y 5 2 p 1 1 } 3 2 x. Compare the graph with the graph
of y 5 1 1 } 3 2 x.
Solution
x y 5 1 1 } 3 2 x y 5 2 p 1 1 } 3 2 x
22 9 18
21 3 6
0 1 2
1 1 } 3 2 }
3
2 1 } 9 2 }
9
x
y
1
3
5
7
9
12123 3
y 5x( )1
3
y 5 2 ? x( )1
3
Because the y-values for y 5 2 p 1 1 } 3 2 x are 2 times
the corresponding y-values for y 5 1 1 } 3 2 x, the graph of
y 5 2 p 1 1 } 3 2 x is a vertical stretch of the graph of
y 5 1 1 } 3 2 x.
Example 2 Compare graphs of exponential functions
1. Graph y 5 22 p 1 1 } 3 2 x. Compare the graph with the
graph of 1 1 } 3 2 x.
x
y
3
9
12123 323
29
215
y 5x( )1
3
y 5 22 ? x( )1
3
The graph of y 5 22 p 1 1 } 3 2 x
is a vertical stretch and a
reflection in the x-axis of
the graph of y 5 1 1 } 3 2 x.
Checkpoint Complete the following exercise.
212 Lesson 8.6 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
Tell whether the graph represents exponential growth or exponential decay. Then write a rule for the function.
SolutionThe graph represents
x
y
1
3
5
12121
3 5
(1, 1)
(0, 5) (y 5 abx where 0 < b < 1). The y-intercept is , so a 5 . Find the value of b by using the point (1, 1) and a 5 .
y 5 abx Write function.
5 p b Substitute.
5 b Solve.
A function rule is .
Example 3 Classify and write rules for functions
EXPONENTIAL GROWTH AND DECAY
Exponential Growth Exponential Decay
y 5 abx, a > 0 y 5 abx, a > 0 and b > 1 and 0 < b < 1
x
y
(0, a)
x
y
(0, a)
Copyright © Holt McDougal. All rights reserved. Lesson 8.6 • Algebra 1 Notetaking Guide 213
EXPONENTIAL DECAY MODEL
y 5 a(1 1 r)t
a is the . r is the .
1 2 r is the . t is the .
Your Notes
Tell whether the graph represents exponential growth or exponential decay. Then write a rule for the function.
SolutionThe graph represents
x
y
1
3
5
12121
3 5
(1, 1)
(0, 5) exponential decay (y 5 abx where 0 < b < 1). The y-intercept is 5 , so a 5 5 . Find the value of b by using the point (1, 1) and a 5 5 .
y 5 abx Write function.
1 5 5 p b 1 Substitute.
0.2 5 b Solve.
A function rule is y 5 5(0.2)x .
Example 3 Classify and write rules for functions
EXPONENTIAL GROWTH AND DECAY
Exponential Growth Exponential Decay
y 5 abx, a > 0 y 5 abx, a > 0 and b > 1 and 0 < b < 1
x
y
(0, a)
x
y
(0, a)
Copyright © Holt McDougal. All rights reserved. Lesson 8.6 • Algebra 1 Notetaking Guide 213
EXPONENTIAL DECAY MODEL
y 5 a(1 1 r)t
a is the initial amount . r is the decay rate .
1 2 r is the decay factor . t is the time period .
Your Notes
Homework
Population The population of a city decreased from 1995 to 2003 by 1.5% annually. In 1995 there were about 357,000 people living in the city. Write a function that models the city's population since 1995. Then find the population in 2003.
SolutionLet P be the population of the city (in thousands), and let t be the time (in years) since 1995. The initial value is , and the decay rate is .
P 5 a(1 2 r)t Write exponential decay model.
5 (1 2 )t Substitute for a, and for r.
5 Simplify.
To find the population in 2003, years after 1995, substitute for t.
P 5 Substitute for t.
ø Use a calculator.
The city's population was about in 2003.
Example 4 Use the exponential decay model
2. The graph of an exponential function passes through the points (0, 4) and (1, 10).
x
y
2
6
10
14
1212322
3
Graph the function. Tell whether the graph represents exponential growth or exponential decay. Then write a rule for the function.
3. In Example 4, suppose that the decay rate of the city's population remains the same beyond 2003. What will be the population in 2020?
Checkpoint Complete the following exercises.
214 Lesson 8.6 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Your Notes
Homework
Population The population of a city decreased from 1995 to 2003 by 1.5% annually. In 1995 there were about 357,000 people living in the city. Write a function that models the city's population since 1995. Then find the population in 2003.
SolutionLet P be the population of the city (in thousands), and let t be the time (in years) since 1995. The initial value is 357 , and the decay rate is 0.015 .
P 5 a(1 2 r)t Write exponential decay model.
5 357 (1 2 0.015 )t Substitute 357 for a, and 0.015 for r.
5 357(0.985)t Simplify.
To find the population in 2003, 8 years after 1995, substitute 8 for t.
P 5 357(0.985)8 Substitute 8 for t.
ø 316.3 Use a calculator.
The city's population was about 316,300 in 2003.
Example 4 Use the exponential decay model
2. The graph of an exponential function passes through the points (0, 4) and (1, 10).
x
y
2
6
10
14
1212322
3
(1, 10)
(0, 4)
Graph the function. Tell whether the graph represents exponential growth or exponential decay. Then write a rule for the function.
Exponential growth;y 5 4(2.5)x
3. In Example 4, suppose that the decay rate of the city's population remains the same beyond 2003. What will be the population in 2020?
about 244,700
Checkpoint Complete the following exercises.
214 Lesson 8.6 • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Graph the sequence 243, 81, 27, 9, 3, …. Let each term’s position number be the x-value. The term is the corresponding y-value.
Position, x 1 2
Term, y 243 27
Example 2 Graph a geometric sequence
Copyright © Holt McDougal. All rights reserved. 8.6 Focus On Functions • Algebra 1 Notetaking Guide 215
Relate Geometric Sequences to Exponential FunctionsGoal p Identify, graph, and write geometric sequences.
VOCABULARY
Geometric sequence
Common ratio
Tell whether the sequence 243, 81, 27, 9, 3, ... is arithmetic or geometric. Then write the next term of the sequence.
Solution
The first term is a1 5 . Find the of consecutive terms:
a2
} a1 5 5
a3 } a2 5 5 5 } 5 5 5
The ratios are . The sequence is . The
common ratio is .
The next term of the sequence is a6 5 p 5 .
Example 1 Identify a geometric sequence
The sequence is a function. The domain is the set of positive numbers (all positive whole numbers). The range is the set of terms.
1 2 3 4 5 6
y
x
200
50
100
150
Your Notes
Focus On FunctionsUse after Lesson 8.6
Graph the sequence 243, 81, 27, 9, 3, …. Let each term’s position number be the x-value. The term is the corresponding y-value.
Position, x 1 2 3 4 5 6Term, y 243 81 27 9 3 1
Example 2 Graph a geometric sequence
Copyright © Holt McDougal. All rights reserved. 8.6 Focus On Functions • Algebra 1 Notetaking Guide 215
Relate Geometric Sequences to Exponential FunctionsGoal p Identify, graph, and write geometric sequences.
VOCABULARY
Geometric sequence Sequence where the ratio of any term to the previous term is constant
Common ratio The constant ratio in a geometric sequence
Tell whether the sequence 243, 81, 27, 9, 3, ... is arithmetic or geometric. Then write the next term of the sequence.
Solution
The first term is a1 5 243. Find the ratios of consecutive terms:
a2
} a1 5 81
} 243 5 1 } 3 a3
} a2 5 27
} 81 5 1 } 3 a4
} a3 5
9 }
27 5 1 } 3
a5 } a4 5
3 } 9 5
1 } 3
The ratios are constant. The sequence is geometric. The
common ratio is 1 } 3 .
The next term of the sequence is a6 5 a5 p 1 } 3 5 1.
Example 1 Identify a geometric sequence
The sequence is a function. The domain is the set of positive numbers (all positive whole numbers). The range is the set of terms.
1 2 3 4 5 6
y
x
200
50
100
150
Your Notes
Focus On FunctionsUse after Lesson 8.6
Homework
Your NotesGENERAL RULE OF A GEOMETRIC SEQUENCE
The nth term of a with first a1 and common r is given by: an 5 .
Write a rule for the nth term of the geometric sequence 243, 81, 27, 9, 3, …. Then find a10 .
Solution
Write the general rule an 5 . Then substitute
values for and to give an 5 p . For the
tenth term, n 5 , so
a10 5 p 5 .
Example 3 Write a rule for a geometric sequence
1. Tell whether the sequence is arithmetic or geometric. Write the next term in the sequence.
7, 221, 63, 2189, …
2. Graph the sequence 7, 221, 63, 2189, …
3. Write a rule for the nth term of the geometric sequence and find a8. 3, 33, 363, 3993, …
Checkpoint Complete the following exercise.
1 2 3 4 5
400
600
200
�200
y
x
216 8.6 Focus On Functions • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Homework
Your NotesGENERAL RULE OF A GEOMETRIC SEQUENCE
The nth term of a geometric sequence with first term a1 and common ratio r is given by: an 5 a1r n21.
Write a rule for the nth term of the geometric sequence 243, 81, 27, 9, 3, …. Then find a10 .
Solution
Write the general rule an 5 a1r n21 . Then substitute
values for a1 and r to give an 5 243 p 1 1 } 3 2 n 21 . For the
tenth term, n 5 10, so
a10 5 243 p 1 1 } 3 2 10 21 5 1 } 81 .
Example 3 Write a rule for a geometric sequence
1. Tell whether the sequence is arithmetic or geometric. Write the next term in the sequence.
7, 221, 63, 2189, …
Geometric; 567
2. Graph the sequence 7, 221, 63, 2189, …
3. Write a rule for the nth term of the geometric sequence and find a8. 3, 33, 363, 3993, …
an 5 3 p 11n21; 58,461,513
Checkpoint Complete the following exercise.
1 2 3 4 5
400
600
200
�200
y
x
216 8.6 Focus On Functions • Algebra 1 Notetaking Guide Copyright © Holt McDougal. All rights reserved.
Words to ReviewGive an example of the vocabulary word.
Review your notes and Chapter 8 by using the Chapter Review on pages 560–563 of your textbook.
Order of magnitude
Exponential function
Compound interest
Geometric sequence
Scientific notation
Exponential growth
Exponential decay
Common ratio
Copyright © Holt McDougal. All rights reserved. Words to Review • Algebra 1 Notetaking Guide 217
Words to ReviewGive an example of the vocabulary word.
Review your notes and Chapter 8 by using the Chapter Review on pages 560–563 of your textbook.
Order of magnitude
The order of magnitude of 96,000,000 is 108.
Exponential function
y 5 3 p 2x
Compound interest
Interest earned on both an initial investment and previously earned interest
Geometric sequence
2, 10, 50, 250, 1250,…
Scientific notation
3.7 3 1025
Exponential growth
y 5 16(2.5)t
Exponential decay
y 5 2.85(0.05)t
Common ratio
3 for the sequence 4, 12, 36, 108,…
Copyright © Holt McDougal. All rights reserved. Words to Review • Algebra 1 Notetaking Guide 217