s3.amazonaws.coms3.amazonaws.com/scschoolfiles/133/algebra_ch_01.doc · web viewthe study of the...

Arithmetic The study of the properties of and relations of numbers, using four basic operationsaddition, subtraction, multiplication, and division

Statement A sentence that is true or false

Algebra The branch of mathematics that uses both letters and numbers to show relations between quantities

Open statement A sentence that is neither true nor false

3n means 3 times n. In algebra, the multiplication sign can be confused with the letter x. This is how you show multiplication in algebra: 3n or 3 • n or 3(n).

You have learned to solve problem with whole numbers. Look at these problems.

3·5 =. 31 + 4 =. 31 - 3 =. 30 -i- 5 =.

If you perform the arithmetic correctly, the statements are true, but if you make a mistake, the statements are false.

True Statements 3·5=15 31 + 4 = 35 31 - 3 = 28 30 -i- 5 = 6

False Statements 3·5=12 31+4=36 31 - 3 = 27 30 --'-- 5 = 4

In algebra, you perform arithmetic that includes letters as well as numbers. For example, if you let n be a placeholder for a number, then

3 times some number is written 3n

31 plus some number is written

31 minus some number is written

31 + n

31 - n

30 --:- nor lQ_ n

30 divided by some number is written

The following statement is a true statement: 3 • 5 = 15. 3n =

15 is neither true nor false. It is an open statement.

Open Statements 3n = 15 31 + n = 35 31 - n = 28 30 --'-- n = 6

2 Chapter 1 Algebra: Arithmetic with Letters

Open statements become true or false statements when you substitute numbers for letters.

Is 3n = 15 true or false when n = 1,2, 3,4, or 57

When n = 1, 3n = 15 becomes 3 • 1 = 15 or 3 = 15 False. When n = 2, 3n = 15 becomes 3 • 2 = 15 or 6 = 15 False. When n = 3, 3n = 15 becomes 3 • 3 = 15 or 9 = 15 False. When n = 4, 3n = 15 becomes 3 • 4 = 15 or 12 = 15 False. When n = 5, 3n = 15 becomes 3 • 5 = 15 or 15 = 15 True.

Exercise A Write true or false for each statement.

1.3·4 = 12

1.6 + 12 = 15

1.19 - 7 = 12

1.14 -;- 7 = 2

6.42 + 8 = 50

6.6 + 18 = 22

6.41 - 5 = 45

6.4·9 = 36

5. ~= 8 10. 84 - 44 = 50

Exercise B Write true, false, or open for each statement.

11.4+6=10 16. 60 + 17 = 67

12. 4n = 16 17. 3

-=1 n

13. 20 -;- 5 = 5 18. 51 + n = 70

14. 60 - n = 50 19. 46 - 16 = 30

15. 60 - 10 = 50 20. 4·6= 24 Exercise C Write true or false for each example. Is

6 + n = 10 true or false when 21. n = 1 22. n = 3 23. n = 4

Is 50 - n = 40 true or false when

27. n = 12 28. n = 4 29. n = 6

24. n = 10 25. n = 5 26. n = 15

Is 24 -;- n = 4 true or false when

30. n = 2

Algebra: Arithmetic with Letters Chapter 1 3

A numerical expression includes only numbers and a one operation. Numerical

expression 9 + 2

7'2 15 - 11

64 --:- 8

A mathematical sentence that includes operations and numbers An algebraic expression includes a variable wit: it

coefficient, if it has one, and at least one operation. Operation The mathematical processes of addition, subtraction, multipli-cation, and division

3y + 4

3 + 2q

m - 17

x--:-6

Algebraic expression Study the table.

S + 11 is a numerical expression. Addi 'on is the operation.

6x + 1 is an algebraic expression and is the variable. The coefficient of x is 6. There are two operations in this expression-m I 'plication and addition.

A mathematical sentence that includes at least one operation and a variable

Variable A letter or symbol that stands for an unknown number

5 + 11 Addi 'Oil

6x + 1 No Yes

Multiplication I x

Addi 'Oil

c";-4 No Yes J c

2'8 Yes No one

Coefficient The number that multiplies the variable

Exercise A Write numerical or algebraic for each expression. Identify the operations.

1. m --:- 8 6. 1.5 - L_

2. 9' 4.5 7. 4 + 6n

3. x + 10 8. 13 • 2

4. 6+9 9. 2'3n

5. 14 - 3 10.

n

-

4 4 Chapter 1 Algebra: Arithmetic with Letters

Exercise B Name the variable in each expression.

11. 6d + 8 15. k-7 19. n -:- 22

12. 5 + m 16. 5+y 20. 5e + 2

13. 18 - e 17. 5y + 5

14. 4h 18. 7

-

x

Exercise C Identify the operation or operations in each expression.

21. 2x 25. 7v -i- 3 29. 4y

22. n-:-4 26. 7 - 2p 30. 6m + 1

23. 3d - 2 27. 6(n)

24. 9 + 5 28.

r

-

7

Exercise D Classify each expression, name the operation or operations, and identify

any variables.

31. 3 + 6 35. 2y + 7 39. 8m + 2

32. 8-4 36. 9-n 40. 9-:-2

33. 3x 37. 11 ·4

34. m-:-6 38. 9 + 3

Exercise E Solve each problem.

41.Write a numerical expression for the number of days in three weeks.

41.There are twenty years in a score. Write a numerical expression to represent the number of years in four scores.

43.Juan was born three years before his brother, who is now 18 years old. Write a numerical expression that represents Juan's age.

43.Write an algebraic expression to represent how far a car travels at 50 mph in d hours.

n.

45.Write an algebraic expression for the following pattern. Use n as the variable.

3 • 1, 3 • 2, 3 • 3, 3 • 4, 3 • 5, 3 • 6, ...


Real number A number on the number line

Integer A whole number or its opposite (' .. -2, -1, 0, 1,2, ... )

Negative integer A whole number less than zero

Positive integer A whole number greater than zero

Opposites Numbers the same distance from zero but on different sides of zero on the number line

Numbers on the number line are examples oi real number. Every point on the number line corre r ~ - to a - -hi - rea number. The arrows at the end of the - umber line how that the pattern of numbers continue.


•••. I I

-1

I 4

-5

-4 -3 -2

o 2

3 5

Positive and negative whole number and zero are called integers. Numbers to the left of zero are negative integers and are read as negative I, negati ·e _ and so on.

•.•. I I I I I -5

-4 -3 -2 -1 o

Numbers to the right of zero are positive integers and are read as positive 1 or I, positive 2 or 2 and so on. Zero is neither negative or positive.

I •. I2

o 3 4

5

For every number other than 0, there is an opposite number on the other side of zero. Opposites are the same distance from zero. When you add two opposites, their sum is zero.

•••• I I I I I I I I I I ••-5 -4 -3 -2 -1 0 2 3 4 5

- 5 is the opposite of 5.

5 is the opposite of - 5.

5 is 5 units from O. -5 is also 5 units from O.

:-so

The distance an integer is from zero on the number line is Absolute value called its absolute value.

I,

The distance a number on the number line is @MB In algebra, Jromzero \-5\ is read "the absolute value of negative 5."

\-5\ = 5, 5 units from O.

\5\ = 5, 5 units from o. I

I i

Exercise A Find the opposite of each integer. i

1.4 5. + II 9. 9

1.-1 6. -17 10. -3

1.6 7. -24

1.-8 8. +14

Exercise 8 Find each absolute value.

11.1-21 15. 1-61 19. 1-81

11.1131 16. 1101 20. 101

11.1+41 17. 1-31 I

11.151 18. I-lll

Exercise C Solve each problem.

21.A football team gained 15 yards on 24. Two different integers are the same

the first play and lost 8 yards on the distance apart on a number line. second. Use + and - numbers to What word could you use to describe . show the team's progress. those integers?

22.How does a temperature of 4°F 25. Explain how you could use a number compare to -7°P line to represent the time when an

23.How does a temperature of - 2°F event happened in the past.

compare to - 6°P

I[:~Y » A bell will sound in exactly two minutes to signal the end of your class. Would you- \y .tilL ~.. use -2 or +2 to describe the number of minutes until the bell rings? Explain. , -- -'


Exercise 8 Find each absolute value.

11. 1-21 15. 1-61 19. 1-81

12. 1131 16. 1101 20. 101

13. 1+41 17. 1-31

14. 151 18. I-lll

read

r

Addition The arithmetic operation of combining numbers to find their sum or total

3 +4=. Start at 3, move 4 units to the right. Since you

stopped at 7, 3 + 4 = 7. +4

·1••••• I I I I I I I I

•I

-1 0 2 3 4 5 6 7 8 9

Addend

A number that is added to one or more numbers

Stun Adding a negative to a positive makes the result less positive. The answer to an

addition problem 3 + (-4) =. Start at 3, move 4 units to the left. Since you stopped at -1,3 + (-4) = -l.

-4 Addition is combining numbers to form a total. Each number being added is an addend. The answer is the sum.

-2 -1 o 2

3 4 5

6

7

8

Adding a positive to a negative makes the result less negative.

-3 + 4 =. Start at -3, move 4 units to the right. Since you

stopped at 1, -3 + 4 = 1.

+4 ·1 ••••• I I I I I I I I I •

I I

-4 -3 -2 -1 0 1 2 3 4 5 6 Adding a negative to a negative makes the result more negative.

-3+(-4)=. Start at -3, move 4 units to the left. Since you stopped at -7, -3 + (-4) = -7.

-4

I I I I -8 -7 -6 -5 -4 -3 -2 -1 o 2


Exercise A Find each sum.

1. S + 8 6. -8 + (-6) 11. -4 + (-2)

2. -9+(-3) 7. -S + 3 12. 3+2

3. -4 + 8 8. 6+9 13. S + (-9)

4. 2 + 7 9. -2 + (-2) 14. -7 + (-4)

5. -10 + 2 10. 6+(-10) 15. 3 + (-3)

Exercise B Find each temperature.

16. -Sop + 4°P 19. 4°P + (_4)OP 22. 6°P + (-lS)OP

17. -8°P + 6°P 20. -lSoP + (_9)OP 23. -Sop + lSoP

18. l30p + 7°P 21. -3°P + lloP 24. 2°P + (-10)OP

Exercise C Pind each temperature.

25. -SoC + (-S)OC 27. -4°C + 10°C 29. 7°C + -18°C

26. 9°C + 7°C 28. 3°C + (-3)OC 30. -2°C + (-9)OC

The E2j key on your calculator changes the sign of the number entered.

You can use the E2j key to add integers.

5 + -1 Press 5 [±] I E2j B '-{ -4 + 8 Press '-{ E2j [±] 8 B '-{

Exercise D Find each sum using a calculator.

31. 6S1 + -821 33. 6S8 + -427 35. -9S1 + 4S8

32. -725 + -26S 34. 326 + 989

Algebra: Arithmetic with Letters 'Chapter 1 9

Subtraction The arithmetic operation of taking one number away from another to find the difference

Difference The answer to a subtraction problem

Subtraction and addition are opposite arithmetic operations. In addition, two (or more) numbers are combined. In subtraction, one number is taken away from another number. The answer is the difference.

Subtracting a positive from a positive makes the result less positive or more negative, so you move to the left.

3 - (+4) = • Start at 3, move 4 units to the left. Since you stopped at -1, 3 - (+4) = -1. Note: 3 - (+4) = -1 gives the same result as 3 + (-4) = -1 because 4 and -4 are opposites.

4

-2 -1 0 2 3 4 5

6 7 8

Subtracting a negative from a negative makes the result less negative or more positive, so you move to the right.

- 3 - (-4) = • Start at - 3, move 4 units to the right. Since you stopped at + 1, - 3 - (-4) = 1. Note: - 3 - (-4) = 1 gives the same result as - 3 + 4 = 1 because -4 and 4 are opposites.

4

• I I I -4 -3 -2 -1 o I

3

I • 2 4 5 6

Subtracting a negative from a positive make it Ie negative or more positive, so you move to the right.

3 - (-4) = • Start at 3, move 4 units to the right. Since you stopped at + 7, 3 - (-4) = 7.

4

o I5

I • -1 9

I 8

2 3 4 6

7

Subtracting a positive from a negative make it le positive or more negative, so you move to the left.

-3 - (+4) = • Start at -3, mo e units to the left. Since you stopped at -7, - 3 - 4 = -7.

4

I I· I I -8 -7 -6 -5 -4 -3 -

2

• I

-1

o 1

2


IN SUMMARY: 3 - (+4) = -1 is the same as 3+(-4)=-1

- 3 - (-4) = 1 is the same as -3 + 4 = 1

3 - (-4) = 7 is the same as 3+4=7

-3 - (+4) = -7 is the same as -3 + (-4) = -7

Rule To subtract in algebra, add the opposite. 3-(+4)=3+(-4) a - (b) = a + (- b)

Exercise A Rewrite each subtraction expression as an addition expression. Solve the new expression.

1. 5 - (+4) 5. -7 - (+5) 9. 8 - (-1)

2. 8-2 6. -3 - 5 10. 4 - 3

3. -5 - (-6) 7. -3-(-10) 11. -6 - (+2)

4. -9 - (-8) 8. 11-(+6) 12. 7 - 3

Exercise B Find each difference.

13. 9-6 18. -7 - (-3) 23. 5 - 8

14. -5 - (-8) 19. 7 - (+9) 24. -7 - (+6)

15. 8-(-8) 20. -6 - 6 25. 6 - (-9)

16. -5-(+10) 21. 3 - 6 26. -10-(-7)

17. 12 - (-3) 22. -3 - (+5) 27. 8 - 2

Exercise C Solve each problem.

eor 28.The record high temperature for Pennsylvania is 111°F. The record low is -42°F. What is the difference between the high and low?

28.What is the difference between Montana's record low of -70°F and New York's record low of - 52°F?

30.Lake Eyre, Australia, has an elevation of - 52 feet, while Lake Torrens, Australia, has an elevation of 92 feet. What is the difference between the elevations?


Multiplication The arithmetic operation of adding a number to itself many times

Factor

A number that is multiplied in a multiplication problem

Product The answer to a multiplication problem

In multiplication, you simply add a number many times. The order in which you multiply two factors does not change the product.

In algebra, "3 times 3" is written as (3)(3) and "3 times n" is written as 3n. You know that (3)(3) = 9. You can think of this as three groups of 3.

(3)(3) = 9

Start at zero and count by 3's on the number line.

·12

./ 3

'/• , , I

, , , •I I

0 2 3 4 5 6 7 8 9 10 Rule (Positive). (Positive) = (Positive)

(3)(-3) =. Start at zero and count by - 3 on the number line. (3)( - 3) means three groups of - 3. (-3) + (-3) + (-3). Therefore, (3)(-3) = -9.

3 2 / 1-, I 1-, ,. , -9 -8 -7 -6 -5 -4 -3 -2 -1 o

Rule (Positive). (Negative) = (Negative)

(-3)(3) =. Treat this the same as (3)( - 3). (- 3)(3) means three groups of (- 3) or (-3) + (-3) + (-3). Therefore, (-3)(3) = -9.

-9 -8 -7 -6 -5 -4 -3 -2 -1 o Rule (Negative). (Positive) = (Negative)


· . , This leaves only one other case, namely (- 3)( - 3) or a (Negative) (Negative). This case cannot be shown on the number line. The product is 9. You need to solve exercises such as these using the following rule:

Rule (Negative)· ( Negative) = (Positive)

So, (- 3)( - 3) = 9

Exercise A Find each product.

] 1. (7)(8) 8. (-4)(13) 15. (-6)(-5)

2. (-4)(-3) 9. (6)(-10) 16. (-8)(-2)

3. (-5)(6) 10. (3)(9) 17. (5)(-10)

4. (9)(-8) 11. (-7)( -9) 18. (15)(4)

5. (9)(9) 12. (-7)(3) 19. (-4)(5)

6. (-5)(-9) 13. (8)(3) 20. (-11)( -8)

7. (5)(12) 14. (-9)(2) ~

Exercise B Tell whether each product is positive, negative, or zero.

] 21.(-34)(-63)

21.(67)( -326)

21.(-487)(-351)

24.(-400)(205)

24.(0)( -345)

24.(800)( -72)

27.(-771)( - 522)

27.(389)(399)

Exercise C Solve these problems.

30. Is (-3)(0) equal to -3 or O? Why?

29.One side of a ship has marks spaced three feet apart. Four marks are underwater. How many feet of the ship are underwater?


Division The arithmetic operation that finds how many times a number is contained in another number

Quotient The answer to a division problem

Dividend A number that is divided

Divisor The number by which you are dividing

L '"

Division is the opposite of multiplication. A dividend is divided by a divisor to find a quotient.

Division is the arithmetic operation that finds how many times a number is contained in another number. The answer is the quotient.

30 -i- 6 = 5

5 _______quotient divisor - 6'3"0

}5U_ dividend

Division and multiplication are opposite operations. Multiplying 3 by 4, then dividing the product by 4 gets you back to 3: (3)(4) = 12 and 12 -7- 4 = 3. You can use this information to discover the

rules for division with negatives.

Multiplication Division

(3)(4) = 12 and 12-'-4=3

Rule (+)(+) = (+) (+) + (+) = (+)

(3)( ~4) = -12 and (~12) -'- (~4) = 3

Rule (+)(-) = H (-) + (-) = (+)

( 3)(4) 12 and ( 12) : (4) ( 3)

Rule (-)(+) = H (-) + (+) = (-)

(~3)(~4)=12 and (12) -i- (~4) = (~3)

Rule (-)(-) = (+) (L)+(-)=H

Rules Like signs create positive products and quotients. Unlike signs create negative products and quotients.


iplying 3)(4) =

'er the

! ! J ! ! ! j ! ! 1 ! ! Writing About Mathematics

Exercise A Find each quotient.

1. 42..;- 6

11.0..;-(-1)

Explain how to 2. -12..;-4 12. 40 -i- (- 5) check division 3. 16 + (-4) 13. 50 ..;- 5 problems using multiplication. 4. -25";-(-5) 14. -40 ..;- (-40) Then write about ways you can use 5. 81 ..;- 9 15. 21 ..;- 3 division and multiplication in 6. -36..;- (-6) 16. 32 -i- (-8) daily activities. 7. -54";-(-9) 17. 64 ..;- 8

8. 48 ..;- 8 18. 21..;- (-3)

9. - 56 ..;- 7 19. -18..;- (-6)

10. -8 -i- (-4) 20. -72 ..;- 9

Exercise B Tell whether each quotient is positive, negative, or zero.

21. 22267 (-42) 25. -85147 (-33)

22. -3458719 26. 1217(-11)

23. 6767 (-26) 27. -356377

24. 54027(73) 28. 07 (-21)

Exercise C Write + or - in each. to make each statement true.

-3) 29 .• 21 73 = 7 30 .• 3073 = -10

Creating Color Want to paint your room? You can choose from thousands of shades of

color. To create each shade, a specified number of drops of one or more base colors is added to white paint. Want to work with color on your computer? Open a computer's drawing or paint program. Go to the "custom color" option where you'll see several cells with numbers. Experiment. If you add or subtract from the numbers in one or more of the cells, you'll see the new color you've created.

- 3)

Algebra: Arithmetic with Letters Chapter 1 15 ---- ~~~-- --~- - --

-- ---- - - -. ..... .

Term Part of an expression separated by an addition or subtraction sign

Like terms Terms that have the same variable

Simplify Combine like terms

In algebra, you need to add, subtract, multiply, and divide using variables to stand for numbers.

In algebra 311 means 11 + n + n. 211 means n + n. 11 mean In (the 1 is not written).

3n and 211 and 11 are all called terms. 3n + 2n is an example of an algebraic expres ion that includes like terms.

To simplify an algebraic ex-pression, combine the like terms.

3n + 2n

Combine the like terms, 3n and 2n. 3n

+ 2n means the same as

n + n - n - n + n = 5n '-.-' ..__.., 3n 2n = 5n T

3x - 15 - lOx

Combine the like terms by subtracting the x's: 3x -

lOx (Think3-10=-7) 3x - lOx = -7x

Since you cannot combine -15 with -lx, you are finished. The simplified answer: 3x - 15 - 10x = -7 x - 15

Exercise A Simplify each expression.

1. m+ m 3. v+ v+ v 5. c+ c+ c+ c 2. 5 + 5 + 5 + 5 4. b + b + b


~e using

rple

rms.

- 15

+c

Exercise B Simplify each expression.

6. 2m+ m 15. 9 + 6c - 2c 24. -St+ t+ (-12t)

7. 6h + 4h 16. 8x + 4 - 3x 25. 14x - (-14x)

8. 3t + 6t 17. 6q - 3 - 2q 26. 6 + j + (-lSj)

9. k + 7k 18. lOp + 12 - 8p 27. 2r + 18 - 18r

10. j + 3j + 6j 19. 20v + 9 + 9v 28. Sf + (-3j) + 14

11. Sp + 6 + 8p 20. 17w- S -12w 29. lSu - 18 - 17u

12. 7 + 2i + 4i 21. -6g+ g 30. 7 + 23m - (-14m)

13. 7y+ 3y- 4 22. k + (-12k)

14. 6z + 4z - 11 23. 6m + (-18m)

Exercise C Find the missing term to make each statement true.

31.c + • = 10c

31.6j -. = 3j

31.18e + • = 12e

34 .• - (-21x) = 21x 35.

90z + • = 80z

Exercise D Write an expression for each statement.

39.Three subtracted from 4p.

39.Twenty-five added to 17 q. 36.The sum of four x and twenty.

36.The difference between 2n and thirty.

38.The sum of Sd and seventeen. ~~

r

~ctivity

Estimate: Will the result of a multiplication or division problem be positive or negative?

(37)( -0.12) = ? (37)( -0.12) -7- 1.5 = ? (37)(-0.12) -7- (-1.5) =? (X)(_X2)(_X) -7- (x3)(-x) = ?

Solution: For multiplication and division, an odd number of negative factors or divisors will give a negative result. An even number of negative factors or divisors will give a positive result.


Unlike terms Terms that have different variables

You may have more than one variable in an expression.

30 + b

\/ unlike terms

You cannot combine terms because a and b are unlike terms.

30 + 0

\/ like terms

30 + a are like terms and can be combined to create 40.

To simplify expressions, combine all like terms.

3x + 1 5 + 6x - 7 - Y

Combine x terms: 3x + 6x = 9x

Combine integers: 15 - 7 = +8

Note: You cannot combine unlike terms 9x, y, and 8, so you are finished.

Rewrite 3x + 15 - 6x - 7 + yas 9x + y + 8.

Exercise A Combine like terms. Simplify each expression.

1. 5x + 3x + 4 + 7b 9. SW-13-4w+ y

2. 14m + 7c+ 4c+ 4m 10. 32j + 14 - 30j + 3h + 2h

3. 16 + 4a - 2a + 7u + 2u 11. Su + 7b + 17 + 2b - 4u

4. 5h - 3h + 14 + 15n 12. 25 + 16a - 22 + 7 d - 15a + 7 d

5. 7y + lOp + 18 + lOp + 17y 13. 9g+ 2 + 16r- 7g+ 15 - l3r

6. 13r+ 25 - 8r+ 14g+ 5g 14. S+j-3+4j+17m

7. m + 10 + 2m + 7t 15. 3m + 7y + 5 - m - 6y

8. 20p - 4 - 12p + 5q + 2q



Exercise B Combine like terms. Simplify each expression.

oecause

16. 7t- 14 - 16t+ ge+ (-14e)

17. 8 - 8e - 8e - 4b - 12b

18. IOn - 15 - 5n + 6 + 2e

19. 3x+ (-17) + 21f+ 3x- (-21j)

20. -19m + 5 - (-m) + e - 7e

21. y - 14 + 30e - 2y + 16e

22. 10 + (-4g) + 10 - 13x - 2g + 5x

23. 20j + 20j - 16m - 16m + 16

24. -25d + 25 - 7 + 155 - 20d 25. 9m - 14 - 6m + 4r + 12r

.an be

Exercise C Tell whether each statement is true or false.

and 8,

26.m + 15e + (-3m) - 4 simplifies to 8me

26.30x + 9 + 9m + 14x - 3m simplifies to 34x + 9 + 6m

26.6 + 4m - 17g + 6m + 3gsimplifies to 6 + 10m - 14g

26.17 + 6y - 10 + m + 7m simplifies to 27 + 6y + 8m

26.-j + 105 - j + 12 - 85 simplifies to -2j + 25 + 12

iTechnoJ.ggy Connection

Software Programs Use Formulas Balancing your checkbook is easy when you use a

software program. Just enter the dollar amounts of your checks and deposits in the right place. The software calculates your balance for you. In the same way, other software programs help businesses to do payroll. They also help insurance companies to figure out premiums, and scientists to calculate the growth of bacteria and viruses. These software programs have one thing in common-they all use formulas with variables!

Exponent 23 is an example of a way to show a number (2) multiplied by itself three times.

Number that tells the times another number is a factor

Base

The 3 is called the exponent. The 2 is called the base.

a' a' a can be written as 03 Exponent _______ Base The number being multiplied; a factor

Power y' y' y' y can be written as y4 Exponent ----Base The product of

mUltiplying any number by itself once or many times

You can multiply and divide by adding or subtracting exponents with the same base.

y2 • y3 = y. y • y. y. y = y5 or y2. y3 = y2 + 3 = y5

To multiply, add exponents with the same base.

y3 -7- y2 = ~ = -w = y

or y3 -7- y2 = y3 - 2 = yl = Y

Rule To divide, subtract exponents with the same base.

Terms such as x2 or x3 have no numerical value until you substitute numbers for x.

If x = 3, x2 = 3 • 3 or 9 and x3 = 3 • 3 • 3 or 27. 3 to the second power is 9. 3 to the third power is 27.


.ed by

u or 27.

Exercise A Tell whether each statement is true or false. If the statement is false, write the

correct solution.

1. 23 = 8

2. x· x· x = x3

3. nr:> mS = mlO 4. 32 = 6

~= bl2 • b4

9. a2s --;- as = as 8 10 !_ = t3 •

tS

6.y3. y3 = 6Y

6.n 10 --;- n2 = n8

Exercise B Simplify each expression.

11. w8• w7 16. Xli --;- xs 21. x3 • x • Xs • X4

12. p4 --;- p3 17. d4• d3 • d 22. r yl6

13. b?» b 18. jI8--;-j6 23. ~

g3

14. clO -i- c2 19. t : t8 • tlO 24. r4 -i- r 15. v7 • v7 20. m20 --;- ml8 25. v2• v2 • v8 • v7

Exercise C Find the value of n to make each statement true.

26. y3 • b4 • y6 = y9bn 31.

W12

--wn

w6-

27. a3 • c4 • 2 . a7 = alOcn 32. X4 • y3 • XS • y2 = X nys e14

p3 • q4 • q7 = pnqll 28. -= elO

33. en

29. t- v6 • t2 = tnv6 34. z" --;- Z8 = z6

30. in --;- ilO = i3s

520

35. -= ss

5n Use the ~ or ~ key on your calculator to compute with exponents.

Find 54.

Press 5 [ZJ l{ El The display will show 625. , . Exercise D Use a calculator to find the value of each expression.

36.254

36.(4.1)2

38.C~)3

38.(0.01)2 Algebra: Arithmetic with Letters Chapter 1 21

8

Perimeter The distance around the outside of a shape

Equilateral triangle A triangle with three equal sides

In the formula for finding the perimeter, or distance around, any equilateral triangle,s stands for the length of a side of the triangle. The letter 5 represents an unknown quantity or variable. To find perimeter, add the length of each side of a figure.

Perimeter = 5 + 5 + 5 = 35

If 5 = 2, then 35 = (3)(2) = 6

If 5 = 15, then 35 = (3)(15) = 45 5

5

Square

5

s

5

Exercise A Use the square for Problems 1-4.

1.What is the formula for finding the perimeter of this square?

1.Find the perimeter of a square, when 5 = 4 cm.

1.Find the perimeter of a square, when 5 = 10 m.

1.What is the length of each side of a square when the perimeter is 36 meters?


)UDd,

itity e

s

- quare? , perimeter

Triangle

b

Exercise B Use the triangle for Problems 5-8.

5.What is the formula for finding the perimeter of this triangle?

5.Find the perimeter, when a = 13 em, b = 10 em, and c =

20 em.

7.Find the perimeter, when a = 20 em, b = 30 em, and c = 40 em.

7.What is the length of side c when the perimeter is 100 m, b =

35 m, and a = 40 m?

Rectangle

w Exercise C Use the rectangle for Problems 9-12.

9.What is the formula for finding the perimeter of this rectangle?

10.Find the perimeter, when I = 15 mm and w = 8 mm.

10.Find the perimeter, when I = 10 em and w = 9 em.

10.What is the width of a rectangle when the perimeter is 14 m and the length is 4 m? Algebra: Arithmetic with Letters Chapter 1 23

Pentagon

s Exercise D Use the pentagon for Problems 13-16.

13.What is the formula for finding the perimeter of this regular pentagon?

14.Find the perimeter, when 5 = 6 m.

14.Find the perimeter, when 5 = 8 cm.

14.What is the length of each side of a regular pentagon when the perimeter is 100 km?

The Pentagon in Arlington, Virginia, has five wedge-shaped sections.


when

Rhombus 5

5

5

5

Exercise E Use the rhombus for Problems 17-2l.

17.What is the formula for finding the perimeter of this rhombus?

17.Find the perimeter, when 5 = 15 km.


17.What is the length of each side of a rhombus when the perimeter is 28 km?

17.A rhombus has the same perimeter formula as what other polygon?

Hexagon

5

5

Exercise F Use the hexagon for Problems 22-25.

22.What is the formula for finding the perimeter of this hexagon?

23.Find the perimeter, when 5 = 10 mm.


23.What is the length of each side of a regular hexagon when the perimeter is 72 m?


Using a Formula

The speed that an airplane travels is calculated by the formula g= a- h. Ground speed (g) equals air speed (a) minus head wind speed (h).

Ground speed (9) g Head wind speed (h) 67 mph

Air speed (a) 200 mph

g = 200 - 67 = 133 The ground speed (g) equals 133 mph.

133 mph 0 67 mph 133 = 0 - 67 a = 133 + 67 = 200 The air speed (0) equals 200 mph.

133 mph

133 = 200 - h h = 200 - 1 33 = 67 The head wind speed (h) equals 67 mph.

200 mph h

Exercise Copy the table. Find each missing value.

1. g 175 mph 53 mph 2. 273 mph 293 mph h

3. 155 mph a 15 mph

4. g 112 mph 27 mph 5. g 305 mph 41 mph

You can find an airplane's speed by using an algebraic formula.


)

ter 1 REVIEW

Write the letter of the correct answer.

1.Find the sum of 15 + (-13).

A 28

B 2 C -2 D

-28

2. Find the difference of -6 - (-7).

A 1 B -1

C 14 D -14

3. Find the quotient of - 25 -;- (- 5).

A 5 C -5

B -20 D -30

4. Simplify the equation 8x + 14 + 14x.

A 28x + 8 C 22x + 14

B 22 + 14x D 28x + 8x

5. Simplify the equation a3 • b4 • a6• b',

6.The perimeter of a regular hexagon is 65. Find the perimeter when 5 = 9 m.

A 54m

B 36m

C 45m D

27m


Chapter 1 REV lEW - continued

Identify each expression as either a numerical or an algebraic expression. Example: 5 + 14 Solution: algebraic

expression 7. 5x - 3 8. 9 + 6 10. P -;- 7 11. 16-;.-4 9. 4y

Find the absolute value, or distance from zero. Example: 1-71 Solution: 1-71 = 7 12. 121 14. 161 13. 1-31 15. 1141 16. 1-251

Find each opposite. Example: -8 Solution: +8 or 8

17. +4 18. -6 19. -l.5 20. 0

Find each sum. Example: -8 + -8 Solution: -8 + -8 = -16

22. -8 + (-10) 23. 14 + 14 24. 7 + (-7) 25. -12 + 20

Find each difference.

Example: -4 - 2 Solution: -4 -2 = -6

26. 3 - 7 27. -3 - (-2) 28. 10 - 4 29. 0 - 5

Find each product. Example: (-4)(4) Solution: (-4)(4) = -16

30. (-5)(-8) 31. (6)(-10) 32. (-9)(6) 33. (10)(0)


_ -i- 4

_0

Find each quotient. Example: 8 -i- - 2

Solution: 8 -i- - 2 = -4

34. 16 -:- (-4) 35. -100 -:- 10 36. 27 -:- 9 37. 0 -:- (-8)

Simplify each expression. Example: -120 + 50 Solution: -120 + 5a = -7a

38. -3m + 16 - (-13m) 43. 128v + 11r - 150v - 3r 48. ( d) ( d2) ( d3

) 39. 15y - 20y + 15 44. X4·X3·X 49. n10 -i- n4

40. -7n + 11 - 3x + 9x + 8n 45. nlO -i- n2 SO. (e) (e7) (e)

41. 1.2k - 3.4k 46. V14 -:- VS 51. F -i- j 42. 9 + 4g- 5 47. (e2) (g) (g4) (e3)

Use a formula to solve the problems. Example: The width of a rectangle is x. Its length is twice the measure of its

width. Write an expression in simplest form to represent the perimeter of the rectangle.

Solution: Let x = length Let 2x = width P = x + x + 2x + 2x p= 6x

54.If the perimeter of a square is 152 em, what is the length of each side?

54.If the perimeter of a regular hexagon is 96 m, what is the length of each side?

52.The perimeter of a square is 45. Find the perimeter when 5 = 12 mm.

52.The perimeter of an equilateral triangle is 35. Find the perimeter, when 5 = 15 km.

When studying for a test, write your own test problems with a partner. Then complete each other's test. Double-check your answers. Algebra: Arithmetic with Letters Chapter 1 29

s3.amazonaws.coms3.amazonaws.com/scschoolfiles/133/algebra_ch_01.doc · web viewthe study of the...

Documents