Pre-Algebra

2-7

Properties of Exponents

Evaluate.

271. 33

2. 4 • 4 • 4 • 4

3. b2 for b = 4

4. n2r for n = 3 and r = 2

256

16

18

Warm Up

Learn to apply the properties of exponents and to evaluate the zero exponent.

The factors of a power, such as 74, can be grouped in different ways. Notice the relationship of the exponents in each product.

7 • 7 • 7 • 7 = 74

(7 • 7 • 7) • 7 = 73 • 71 = 74

(7 • 7) • (7 • 7) = 72 • 72 = 74

Words Numbers Algebra

To multiply powers with the same base, keep the base and add the exponents.

bm • bn = bm + n

35 • 38 = 35 + 8 = 313

MULTIPLYING POWERS WITH THE SAME BASE

A. 66 • 63

69

66 + 3

B. n5 • n7

n12

n5 + 7

Add exponents.

Add exponents.

Multiply. Write the product as one power.

Example 1A & 1B: Multiplying Powers with the Same Base

D. 244 • 244

C. 25 • 2

2 6

25 + 1

248

24 4 + 4

Think: 2 = 2 1

Multiply. Write the product as one power.

Add exponents.

Add exponents.

Example 1: Multiplying Powers with the Same Base Continued

A. 42 • 44

46

42 + 4

B. x2 • x3

x5

x2 + 3

Add exponents.

Add exponents.

Multiply. Write the product as one power.

Try This

D. 412 • 417

C. x5 • y2

419

41 2 + 7

Multiply. Write the product as one power.

Cannot combine; the bases are not the same.

Add exponents.

x5 • y2

Try This

Notice what occurs when you divide powers with the same base.

DIVIDING POWERS WITH THE SAME BASE

Words Numbers Algebra

To divide powers with the same base, keep the base and subtract the exponents.

6569 – 469

64= = bm – nbm

bn=

55

53=

5 5 55 5 5 5 5

= 5 • 5 = 52=5 5 5

5 5 5 5 5

Subtract exponents.

72

75 – 3

75

73

Divide. Write the product as one power.

A.

x10

x9B.

Subtract exponents.x10 – 9

x Think: x = x1

Example 2: Dividing Powers with the Same Base

Subtract exponents.

97

99 – 2

99

92

Divide. Write the product as one power.

A.

B. e10

e5

Subtract exponents.e10 – 5

e5

Try This

When the numerator and denominator have the same base and exponent, subtracting the exponents results in a 0 exponent.

This result can be confirmed by writing out the factors.

1 = 42

42 42 – 2 = 40 = 1=

=(4 • 4)(4 • 4) = 11

1 =42

2= (4 • 4)

4 (4 • 4)

00 does not exist because 00 represents a quotient of the form

But the denominator of this quotient is 0, which is impossible, since you cannot divide by 0.

Helpful Hint

0n

0n.

THE ZERO POWER

Words Numbers Algebra

The zero power of any number except 0 equals 1.

1000 = 1

(–7)0 = 1a = 1, if a 0

A light-year, or the distance light travels in one year, is almost 1018 centimeters. To convert this number to kilometers, you must divide by 105. How many kilometers is a light-year?

10 18 - 5

A light-year is almost 1013 km.

10

13

10 18

10 5

Subtract exponents.

Example 3: Astronomy Application

A ship has 107 kilograms of grain loaded into its cargo hold. A metric ton is 103 kilograms. How many metric tons of grain were loaded?

10 7 - 3

The ship had 104 metric tons of grain loaded.

10

4

10 7

10 3

Subtract exponents.

The weight in metric tons is equal to the weight in kilograms divided by 10 kilograms per metric ton.3

Try This

Write the product or quotient as one power

3.

8 9

n 71. n3 n4

109

105 10 4

4.

t 2

5. 33 • 32 • 35 3 10

2. 8 • 88

t9

t7

Lesson Quiz: Part 1

A school would like to purchase new globes. They can get six dozen for $705.80 from Company A. From Company B, they can buy a half gross for $725.10. Which company should they buy from? (1 gross = 122 items)

Company A

6.

Lesson Quiz: Part 2