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©Curriculum Associates, LLC Copying is not permitted. 3Lesson 1 Properties of Integer Exponents

Name: Properties of Integer Exponents

Lesson 1

Vocabularybase the number being

used as a factor in an

exponential expression

5 is the base 53

exponent the number

that shows how many

times a base is used as a

factor

53 3 is the exponent

Prerequisite: Evaluate Numerical Exponential Expressions

Study the example problem showing how to write and evaluate expressions with exponents. Then solve problems 1–9.

1 Look at the table How many times greater is the amount in Jacob’s savings in Week 3 than in Week 2?

2 How much will Jacob have in his account in Week 3?

3 Jacob thinks that 35 is 5 ? 5 ? 5, or 125 Explain what Jacob is doing wrong

4 Margo’s dad off ers to give her 5¢ on Sunday Then for each day of the week, he off ers to give her 5 times the amount from the previous day How much will he give her on Saturday? Write an expression to show how much Margo’s dad gives her on Saturday

Example

Jacob decides to save money for a new tablet He will save $3 the first week and then triple the amount he has saved each week for 5 weeks Write and evaluate an exponential expression to find how much money Jacob will have in his savings in Week 5

Represent the problem with repeated multiplication and exponential expressions

Week 1 Week 2 Week 3 Week 4 Week 5

3 5 31 3 ? 3 5 32 3 ? 3 ? 3 5 33 3 ? 3 ? 3 ? 3 5 34 3 ? 3 ? 3 ? 3 ? 3 5 35

Week 5 expression: 35

Evaluate the expression: 35 5 3 ? 3 ? 3 ? 3 ? 3 5 243

Jacob will have $243 in his savings in Week 5

33

B

B

M

M

three times greater

$27

Jacob is confusing the base and the exponent.

57 5 78,125¢ or $781.25

©Curriculum Associates, LLC Copying is not permitted.4 Lesson 1 Properties of Integer Exponents

Solve.

5 Is 24 equal to 2 ? 4? Explain

6 A bacterium cell splits into 2 cells every hour Write and evaluate an exponential expression to fi nd how many cells there will be in 6 hours Then use your answer to help you fi nd the number of hours it will take for there to be 1,024 cells

Show your work.

Solution:

7 The population of California is about 39 million Is this greater than or less than 107? Explain

8 Write each of the numbers 1, 8, 27, 64, and 125 as a base raised to the third power

1 5 u3 8 5 u3 27 5 u3

64 5 u3 125 5 u3

9 The exponential expression 28 has a value of 256 Write two other exponential expressions that have a value of 256 Explain how you got your answers (Begin by writing out 28 as the product of 2s )

4

B

M

M

M

C

No; 24 means 2 used as a factor 4 times, or 2 ? 2 ? 2 ? 2 5 16, but 2 ? 4 5 8.

Greater than; 107 5 10,000,000 and 39,000,000 10,000,000.

Possible answers: 44 and 162. Possible explanation: I know that 2 ? 2 ? 2 ? 2 ? 2 ? 2 ? 2 ? 2 5 256.

If I put the 2s in groups of 2, I get (2 ? 2) ? (2 ? 2) ? (2 ? 2) ? (2 ? 2) or 4 ? 4 ? 4 ? 4 5 256. If I put the

4s in groups of 2, I get (4 ? 4) ? (4 ? 4) or 16 ? 16 5 256.

Possible work: 6 hours: 26 5 2 ? 2 ? 2 ? 2 ? 2 ? 2 5 64

1,024 cells: 26 5 64, 27 5 64 ? 2 5 128, 28 5 128 ? 2 5 256, 29 5 256 ? 2 5 512, 210 5 512 ? 2 5 1,024

In 6 hours, there will be 64 cells. There will be 1,024 cells in 10 hours.

1 2 3

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Unit 1

Practice Lesson 1 Properties of Integer Exponents

©Curriculum Associates, LLC Copying is not permitted. 5Lesson 1 Properties of Integer Exponents

Name: Lesson 1

Products of Powers

Study the example problems showing how to find the power of a power and the products of powers with the same exponent. Then solve problems 1–10.

1 The expression (82)3 in the example problem is a product of powers What are the powers being multiplied? What are the powers being multiplied in the expression (24)(54)?

2 Simplify: (75)6 Write your answer using an exponent

3 Simplify: (63)(93) Write your answer using an exponent

4 Is the statement (35)4 5 (34)5 true? Explain your reasoning

Example

Same Base and Same Exponent

Simplify: (82)3

One Way:(82)3 5 82 ? 82 ? 82

5 8 ? 8 ? 8 ? 8 ? 8 ? 8 5 86

Another Way:(82)3 5 82 ? 82 ? 82

5 82?3 Multiply the exponents 5 86

Different Base and Same Exponent

Simplify: (24)(54)

One Way:(24)(54) 5 (2 ? 2 ? 2 ? 2)(5 ? 5 ? 5 ? 5) 5 (2 ? 5)(2 ? 5)(2 ? 5)(2 ? 5) 5 10 ? 10 ? 10 ? 10 5 104

Another Way:(24)(54) 5 (2 ? 5)4 Multiply the bases 5 104

5

B

B

B

M

The powers being multiplied in (82)3 are 82, 82, and 82. The powers being multiplied in

(24)(54) are 24 and 54.

Yes; Possible explanation: (35)4 5 35 ? 35 ? 35 ? 35 5 35?4 5 320; (34)5 5 34 ? 34 ? 34 ? 34 ? 34 5

34?5 5 320; Both expressions equal 320.

730

543

©Curriculum Associates, LLC Copying is not permitted.6 Lesson 1 Properties of Integer Exponents

Solve.

5 Simplify: (75)(45) Write your answer using an exponent

6 Explain in words how to simplify: (1532)7

7 Is the statement (105)(45) 5 145 true? Explain your reasoning

8 What is the value of x in the equation (5x)5 5 535? Explain

9 Without evaluating the expressions, tell which is greater, (44)(54) or (25)(105) Explain your reasoning

10 Nicholas says that (26)(26) equals 212 and also equals 46 Do you agree? Explain your reasoning

6

B

B

M

M

M

C

285

Possible explanation: Keep the base, 153, and multiply the exponents, 2 and 7:

(1532)7 5 15314

No; Possible explanation: (105)(45) 5 10 ? 10 ? 10 ? 10 ? 10 ? 4 ? 4 ? 4 ? 4 ? 4 5 (10 ? 4)5 5 405.

When the powers have different bases and the same exponent, the bases should be

multiplied, not added.

7; Possible explanation: The exponent of the simplified expression, 535, is the product of

the exponents of the factors, 5 and x. Because 5 ? 7 5 35, x 5 7.

(25)(105); Possible explanation: The bases are equal because 4 ? 5 5 2 ? 10, but the exponent

for (25)(105), 5, is greater than the exponent for (44)(54), 4. So, (25)(105) (44)(54).

Yes; Possible explanation: Because 4 5 2 ? 2, 46 5 (2 ? 2)(2 ? 2)(2 ? 2)(2 ? 2)(2 ? 2)(2 ? 2), or 212.

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©Curriculum Associates, LLC Copying is not permitted. 7Lesson 1 Properties of Integer Exponents

Name: Lesson 1

Zero and Negative Exponents

Study the example problems showing how to simplify expressions with zero and negative exponents. Then solve problems 1–12.

1 Simplify: 1000

2 Write (–32)22 as an expression without a negative

exponent

3 Write 1 ·· 76 as an expression with a negative exponent

4 Compare 80 and 822 Which is greater? Explain your reasoning

5 Simplify (120)(126) Write your answer using an exponent Explain how you found your answer

6 What is the value of x in the equation (–35)x 5 1? Explain

7 Write an expression using exponents that is equivalent to each of the following expressions

a 523 5

b (216)22 5

c 1 ·· 84 5

Example

Zero Exponent

Simplify: 2170

In general, n0 5 1, where n Þ 0

So, 2170 5 1

Negative Exponent

Simplify: 1522

In general, n2a 5 1 ·· na , where n Þ 0

So, 1522 5 1 ··· 152

7

B

B

B

M

M

M

M

1 ······ (232)2

726

1

80 is greater. Possible explanation: 80 5 1 and 822 5 1 ·· 82 , which is less than 1.

126; Possible explanation: 120 5 1 and 1 ? 126 5 126.

0; Any nonzero number to the power of 0 equals 1.

1 ·· 53

1 ····· (–16)2

8–4

©Curriculum Associates, LLC Copying is not permitted.8 Lesson 1 Properties of Integer Exponents

Solve.

8 Simplify the expression 5225 Then write it as repeated multiplication

9 Write an exponential expression that is equivalent to (414)22

10 Is 622 positive or negative? Explain

11 Write 90, 93, and 922 in order from least to greatest

Show your work.

Solution:

12 Lizbeth says that (–12)23 equals a positive number because the product of two negative integers is a positive integer Do you agree? Use what you know about exponential expressions to explain

8

B

M

M

M

C

1 ···· 525 5 1 ··· 52 ? 1 ··· 52 ? 1 ··· 52 ? 1 ··· 52 ? 1 ··· 52

Possible answers: 4228, 1 ····· (414)2 , or 1 ··· 428

positive; Possible answer: 622 5 1 ·· 62 5 1 ··· 36 , which is positive.

No; Possible explanation: To simplify an exponential expression, you do not multiply the

exponent and the base. (212)23 5 1 ······· (212)3 5 1 ···· 212 ? 1 ···· 212 ? 1 ···· 212 , and the product of 3 negative

numbers is a negative number.

922, 90, 93

90 5 1

93 is greater than 1.

922 5 1 ·· 92 , which is less than 1.

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Unit 1

Practice Lesson 1 Properties of Integer Exponents

©Curriculum Associates, LLC Copying is not permitted.10 Lesson 1 Properties of Integer Exponents

6 Which expression is equivalent to (34 ? 54)23?

A 1 ··· 155

B 15248

C 1 ···· 1512

D 155

Tania chose B as the correct answer How did she get that answer?

Solve.

4 Simplify: 3221 ···· 326 Write your answer with a

positive exponent

Show your work.

Solution:

5 Write 96 as a power with a base of 3

What are the factors of 9?

Remember the order of operations. Simplify the expression within the parentheses first.

The expression is a quotient of powers.

10

B

M

C

Possible answer:

3221 ···· 326 5 322126

5 3227

5 1 ··· 327

96 5 (32)6 5 312

Possible answer: Tania did not use the order of

operations or the rules for working with

exponents. She multiplied the bases and

multiplied all the exponents.

3221 ···· 326 5 1 ··· 327

©Curriculum Associates, LLC Copying is not permitted. 9Lesson 1 Properties of Integer Exponents

Name: Lesson 1

1 Look at the equations below Tell whether each equation is True or False

a 35 ? 375 335 u True u False

b (63 ? 33)2 5 186 u True u False

c 726 ? 1 ·· 74 5 1 ··· 710

u True u False

d 44 ? 42 5 46 u True u False

e 1324 ···· 134 5 130

u True u False

f (23 ? 83)0 5 169 u True u False

Simplify Expressions with Exponents

Solve the problems.

3 Complete the table

Expression 104 ? 1022 54 ? 74 (27 ? 47)3

Simplified Expression

2 Tyler simplifi ed the expression 54 ? 529 All of his work except his answer is shown below

54 ? 529 5 541(29)

5 525

5 ?

Which expression is the correct answer for Tyler’s work?

A 55

B 1 ·· 55

C 1 ··· 525

D 5

You may have to apply more than one rule when working with exponents.

Remember what you know about negative exponents.

Are the bases equal? Are the exponents equal?

9

M

B

B

3333

33

102 354 821