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Study Guide ReviewStudy Guide ReviewGenerating Equivalent Numerical Expressions

How can you generate equivalent numerical expressions and use them to solve real-world problems?

EXAMPLE 1Find the value of each power.

A. 0.92

0.92 = 0.9 × 0.9 = 0.81

B. 180

Any number raised to the power of 0 is 1.

180 = 1

C. ( 1 _ 4 ) 4

( 1 _ 4 ) 4 =  ( 1 _ 4 ) ( 1 _ 4 ) ( 1 _ 4 ) ( 1 _ 4 ) = 1 ___ 256

EXAMPLE 2Find the prime factorization of 60.

MODULE 999

? ESSENTIAL QUESTION

60 = 2 × 2 × 3 × 5

60 = 22 × 3 × 5

The prime factorization of 60 is 22 × 3 × 5.

EXAMPLE 3Simplify each expression.

A. 4 × (23 + 5)

= 4 × (8 + 5)

= 4 × 13

= 52

B. 27 ÷ 32 × 6

= 27 ÷ 9 × 6

= 3 × 6

= 18

EXERCISESUse an exponent to write each expression. (Lesson 9.1)

1. 3.6 × 3.6 2. 9 × 9 × 9 × 9 3. 4 _ 5 ×  4 _ 5 ×  4 _ 5

Key Vocabularybase (base (en

numeración))

exponent (exponente)

order of operations (orden de las operaciones)

power (potencia)

UNIT 4

23 = 8

Add.

Multiply.

32 = 9

Divide.

Multiply.

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285Unit 4

Find the value of each power. (Lesson 9.1)

4. 120 5. 132 6. ( 2 _ 7 ) 3

7. 0.42 8. ( 4 _ 9

) 1 9. 0.73

Find the prime factorization of each number. (Lesson 9.2)

10. 75 11. 29 12. 168

13. Eduardo is building a sandbox that has an area of 84 square feet. What are the possible whole number measurements for the length and width of the sandbox? (Lesson 9.2)

Simplify each expression. (Lesson 9.3)

14. 2 × 52 – (4 + 1) 15. 22 - (32 + 4) __________ 12 ÷ 4

Generating Equivalent Algebraic Expressions

How can you generate equivalent algebraic expressions and use them to solve real-world problems?

EXAMPLE 1Evaluate each expression for the given values of the variables.

A. 2(x2 - 9); x = 5

2(52 - 9)

= 2(16)

= 32

When x = 5, 2(x2 – 9) = 32.

B. w + y 2 + 3w; w = 2, y = 6

2 + 62 + 3(2)

= 2 + 36 + 6

= 44

When w = 2 and y = 6, w + y 2 + 3w = 44.

EXAMPLE 2Determine whether the algebraic expressions are equivalent: 5(x + 2) and 10 + 5x.

5(x + 2) = 5x + 10

= 10 + 5x

5(x + 2) is equal to 10 + 5x. They are equivalent expressions.

MODULE 1010

? ESSENTIAL QUESTION

Key Vocabularyalgebraic expression

(expresión algebraica)

coeffi cients (coefi ciente)

constant (contante)equivalent expressions

(expresión equivalente)

evaluating (evaluar)

term (término (en una expresión))

52 = 25

Subtract.

Multiply.

Distributive Property

Commutative Property

62 = 36

Multiply.

Add.

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Unit 4286

EXERCISESWrite each phrase as an algebraic expression. (Lesson 10.1)

1. x subtracted from 15 2. 12 divided by t

3. 4 groups of y 4. the sum of z and 7

Write a phrase for each algebraic expression. (Lesson 10.1)

5. 8p

6. s + 7

Evaluate each expression for the given values of the variables. (Lesson 10.2)

7. 8z + 3; z = 8 8. 3(7 + x2); x = 2

9. s - 5t + s2; s = 4, t = -1 10. x - y3; x = -7, y = 3

11. The expression 1 _ 2 (h)(b1 + b2) gives the area of a trapezoid, with b1 and b2 representing the two base lengths of a trapezoid and h representing the height. Find the area of a trapezoid with base lengths 4 in. and 6 in. and a height of 8 in. (Lesson 10.2)

Determine if the expressions are equivalent. (Lesson 10.3)

12. 7 + 7x; 7 (x + 1 _ 7 )

13. 2.5(3 + x); 2.5x + 7.5

Combine like terms. (Lesson 10.3)

14. 3m - 6 + m2 - 5m + 1

15. 7x + 4(2x - 6)

16. b2 + 3 + 2b2 + 4 - 7

17. 3(p + 5) - 8 + 11p

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287Unit 4

Personal Math Trainer

Online Assessment and

Interventionmy.hrw.com

UNIT 4 MIXED REVIEW

Selected Response

1. Which expression is equivalent to 2.3 × 2.3 × 2.3 × 2.3 × 2.3?

A 2.3 × 5

B 235

C 25 × 35

D 2.35

2. Which operation should you perform first when you simplify 63 – (2 + 54 × 6) ÷ 5?

A addition

B division

C multiplication

D subtraction

3. Sheena was organizing items in a scrapbook. She took 25 photos and divided them evenly between p pages. Which algebraic expression represents the number of photos on each page?

A p – 25

B 25 – p

C p

__ 25

D 25 __ p

4. Which is another way to write 7 × 7 × 7 × 7?

A 74

B 7(4)

C 28

D 47

5. Angela earns x dollars an hour. On Friday, she worked 6 hours. On Saturday, she worked 8 hours. Which expression shows how much she earned both days?

A 6x + 8

B 8x × 6x

C (6 + 8)x

D 6 + 8 ____ x

6. Marcus is doing a science experiment in which he measures the rate at which bacteria multiply. Every 15 seconds, the bacteria double in number. If there are 10 bacteria now, how many will there be in 2 minutes?

A 160 bacteria

B 256 bacteria

C 1,280 bacteria

D 2,560 bacteria

7. The prime factorization of which number is 25 × 5?

A 50

B 125

C 160

D 500

8. Which expression has a value of 36 when x = 4 and y = 7?

A 2xy

B 2x + 4y

C 6y - x

D 12x - 2y

Assessment Readiness

D

D

C

B

C

D

A

C

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289Unit 4

9. What should you do first to simplify the expression ( 4 3 + 9) ÷ 76 + 5?

A Add 4 and 9.

B Add 76 and 5.

C Multiply 4 × 4 × 4.

D Divide ( 4 3 + 9) by 76.

10. Which ratio is equivalent to 4:10?

A 2 _ 5

B 8 __ 10

C 12 __ 16

D 16 __ 10

11. Travis and Paula went to lunch. Travis ordered a sandwich for $7.50, and Paula ordered a burger for $5.25. After lunch, they left a 15% tip for their waiter. How much money did they spend altogether?

A $12.75

B $14.66

C $15.95

D $16.00

12. Which shows the following numbers in order from greatest to least?

1.5, 2 _ 4 , 4 _ 2 , 1.05

A 4 _ 2 , 1.5, 1.05, 2 _ 4

B 1.05, 1.5, 2 _ 4 , 4 _ 2

C 4 _ 2 , 2 _ 4 , 1.5, 1.05

D 1.05, 4 _ 2 , 2 _ 4 , 1.5

Mini-Tasks

13. For every bag of trail mix the local Scout Guide troop sells, they earn $0.45.

a. Write an expression to represent this situation.

b. Sarah sold 52 bags of trail mix. How much did she earn for her troop?

c. Let x represent the total number of bags of trail mix sold by Sarah’s troop. Write an expression to show what percentage of bags Sarah sold.

14. Robert is replacing sod in two square-shaped areas of his backyard. One side of the first area is 7.5 feet. One side of the other area is 5.7 feet. The sod costs y dollars per square foot.

a. Write an expression to show how much Robert will spend on sod.

b. If the sod costs $3.25 per square foot, about how much will Robert spend to put sod down in both areas of his backyard? Round to the nearest dollar.

15. Jose wants to find how many gallons of water he needs to fill his cube-shaped aquarium. One side of his aquarium is 4 feet long.

a. Write and solve an expression to find the volume of Jose’s aquarium.

b. One cubic foot is equal to 7.48 gallons of water. How many gallons of water does Jose need to fill his aquarium? Round to the nearest gallon.

HotHotTip!Tip!

When possible, use logic to eliminate at least two answer choices.

A

C

B

A

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