1 Lesson 5.2.1 Negative and Zero Exponents Negative and Zero Exponents

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Lesson 5.2.1Lesson 5.2.1

Negative and Zero

Exponents

Negative and Zero

Exponents

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Lesson

5.2.1Negative and Zero ExponentsNegative and Zero Exponents

California Standards:Number Sense 2.1

Understand negative whole-number exponents. Multiply and divide expressions using exponents with a common base.

Algebra and Functions 2.1Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents.

What it means for you:You’ll learn what zero and negative powers mean, and simplify expressions involving them.

Key words:• base• exponent• power

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Negative and Zero ExponentsNegative and Zero ExponentsLesson

5.2.1

Up to now you’ve worked with only positive whole-number exponents. These show the number of times a base is multiplied. As you’ve seen, they follow certain rules and patterns.

The effects of negative and zero exponents are trickier to picture. But you can make sense of them because they follow the same rules and patterns as positive exponents.

20 100 100

831212 9163256 84122

4–32–6 17–26–7 4–102–256

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Negative and Zero ExponentsNegative and Zero Exponents

Any Number Raised to the Power 0 is 1

Lesson

5.2.1

Any number that has an exponent of 0 is equal to 1.

So, 20 = 1, 30 = 1, 100 = 1, = 1.1

2

0

For any number a 0, a0 = 1

You can show this using the division of powers rule.

5

If you start with 1000, and keep dividing by 10, you get this pattern:


5.2.1

1000 = 103

100 = 102

10 = 101

1 = 100

Now divide by 10: 103 ÷ 101 = 10(3 – 1) = 102

Now divide by 10: 102 ÷ 101 = 10(2 – 1) = 101

Now divide by 10: 101 ÷ 101 = 10(1 – 1) = 100

The most important row is the second to last one.

1000 = 103

100 = 102

10 = 101

1 = 100

Now divide by 10: 103 ÷ 101 = 10(3 – 1) = 102

Now divide by 10: 102 ÷ 101 = 10(2 – 1) = 101

Now divide by 10: 101 ÷ 101 = 10(1 – 1) = 100

You also know that 10 divided by 10 is 1. So you can see that 100 = 1.

When you divide 10 by 10, you have 101 ÷ 101 = 10(1 – 1) = 100.

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This pattern works for any base.


5.2.1

For instance,

61 ÷ 61 = 6(1 – 1) = 60, and 6 divided by 6 is 1. So 60 = 1.

You can use the fact that any number to the power 0 is 1 to simplify expressions.

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Example 1

Solution follows…

Lesson

5.2.1

Simplify 34 × 30. Leave your answer in base and exponent form.

Solution

34 × 30 = 34 × 1 = 34

You can use the multiplication of powers rule to show this is right:

Add the exponents of the powers34 × 30 = 3(4 + 0) = 34

You can see that being multiplied by 30 didn’t change 34.

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Guided Practice

Solution follows…

Lesson

5.2.1

Evaluate the following.

1. 40 2. x0 (x 0)

3. 110 + 120 4. (7 + 6)0

5. 43 ÷ 43 6. y2 ÷ y2 (y 0)

7. 32 × 30 8. 24 × 20

9. a8 ÷ a0 (a 0)

1 1

2 1

1 1

32 or 9 24 or 16

a8

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You Can Justify Negative Exponents in the Same Way

Lesson

5.2.1

By continuing the pattern of powers shown below you can begin to understand the meaning of negative exponents.

1000 = 103

100 = 102

10 = 101

1 = 100

Now divide by 10: 103 ÷ 101 = 10(3 – 1) = 102

Now divide by 10: 102 ÷ 101 = 10(2 – 1) = 101

Now divide by 10: 101 ÷ 101 = 10(1 – 1) = 100

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5.2.1

Carry on dividing each power of 10 by 10:

Now divide by 10: 102 ÷ 101 = 10(2 – 1) = 101

Now divide by 10: 101 ÷ 101 = 10(1 – 1) = 100

100 = 102

10 = 101

1 = 100

Now divide by 10: 10–2 ÷ 101 = 10(–2 – 1) = 10–3

Now divide by 10: 10–1 ÷ 101 = 10(–1 – 1) = 10–2

Now divide by 10: 100 ÷ 101 = 10(0 – 1) = 10–1

1

1000 = 10–3

1

100 = 10–2

1

10 = 10–1

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5.2.1

Look at the last rows, shown in red, to see the pattern:

Now divide by 10: 10–2 ÷ 101 = 10(–2 – 1) = 10–3

Now divide by 10: 10–1 ÷ 101 = 10(–1 – 1) = 10–2

1

1000 = 10–3

1

100 = 10–2

1

10 = 10–1

One-tenth, which is , can be rewritten as = 10–1.1

10

1

101One-hundredth, which is , can be rewritten as = 10–2.

1

100

1

102One-thousandth, which is , can be rewritten as = 10–3.

1

1000

1

103

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This works with any number, not just with 10.

Lesson

5.2.1

60 ÷ 61 = 6–1 and 1 ÷ 6 = , so 6–1 = .1

6

1

6

6–1 ÷ 61 = 6–2 and ÷ 6 = • = = , so 6–2 = .1

6

1

6

1

6

1

62

1

36

For example:

60 = 1

This pattern illustrates the general definition for negative exponents.

For any number a 0, a–n = 1

an

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Example 2

Solution follows…

Lesson

5.2.1

Rewrite 5–3 without a negative exponent.

Solution

5–3 = Using the definition of negative exponents1

53

=1

125

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Example 3

Solution follows…

Lesson

5.2.1

Rewrite using a negative exponent.1

75

Solution

= 7–5 Using the definition of negative exponents1

75

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Guided Practice

Solution follows…

Lesson

5.2.1

Rewrite each of the following without a negative exponent.

10. 7–3

11. 5–m

12. x–2 (x 0)

1

73

1

5m

1

x2

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Guided Practice

Solution follows…

Lesson

5.2.1

Rewrite each of the following using a negative exponent.

13.

14.

15. (q 0)

1

33

1

64

1

q × q × q

3–3

6–4

q–3

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Independent Practice

Solution follows…

Lesson

5.2.1

Evaluate the expressions in Exercises 1–3.

1. 87020

2. g0 (g 0)

3. 20 – 30

1

1

0

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Solution follows…

Lesson

5.2.1

Write the expressions in Exercises 4–6 without negative exponents.

4. 45–1

5. x–6 (x 0)

6. y–3 – z–3 (y 0, z 0)

1

45

1

x6

1

y3

1

z3–

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Solution follows…

Lesson

5.2.1

Write the expressions in Exercises 7–9 using negative exponents.

7.

8. (r 0)

9. (p + q 0)

1

82

1

r6

1

(p + q)v

8–2

r–6

(p + q)–v

20



Solution follows…

Lesson

5.2.1

In Exercises 10–12, simplify the expression given.

10. 54 × 50

11. c5 × c0 (c 0)

12. f 3 ÷ f 0 (f 0)

54

c5

f 3

21



Solution follows…

Lesson

5.2.1

13. The number of bacteria in a Petri dish doubles every hour. The numbers of bacteria after each hour are 1, 2, 4, 8, 16, ... Rewrite these numbers as powers of 2.

20, 21, 22, 23, 24

20, 2–1, 2–2, and 2–3

1

2

1

4

1

814. Rewrite the numbers 1, , , and as powers of 2.

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Round UpRound Up

Lesson

5.2.1

So remember — any number (except 0) to the power of 0 is equal to 1. This is useful when you’re simplifying expressions and equations.

Later in this Section, you’ll see how negative powers are used in scientific notation for writing very small numbers efficiently.

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1 Lesson 5.2.1 Negative and Zero Exponents Negative and Zero Exponents