Pre-Algebra

2-7 Properties of ExponentsProperties of Exponents 8-3/8-4/8-5

Pre-Algebra

Multiply Powers

Dividing Powers

Raising a Power to a Power

Pre-Algebra

2-7 Properties of Exponents

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

Pre-Algebra

2-7 Properties of Exponents

Words Numbers Algebra

To raise a power to a power, multiply the exponents.

(bm)n = bmn(35)2 = 310

= 59,049

RAISING A POWER TO A POWER

Pre-Algebra

2-7 Properties of Exponents

Additional Example 1A & 1B: 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.

Pre-Algebra

2-7 Properties of Exponents

D. 244 • 244

C. 25 • 2

26

25 + 1

248

24 4 + 4

Think: 2 = 2 1

Additional Example 1: Multiplying Powers with the Same Base Continued

Multiply. Write the product as one power.

Add exponents.

Add exponents.

Pre-Algebra

2-7 Properties of Exponents

Try This: Example 1A & 1B

A. ( 42 )4

65,536

48

B. ( x2 )3

x6

Simplify.

Pre-Algebra

2-7 Properties of Exponents

D. ( 412 )7

C. x5 • y2

4114

Try This: Example 1C & 1D

Multiply. Write the product as one power.

Cannot combine; the bases are not the same.

x5 • y2

Pre-Algebra

2-7 Properties of Exponents

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

Pre-Algebra

2-7 Properties of Exponents

Subtract exponents.

72

75 – 3

75

73

Additional Example 2: Dividing Powers with the Same Base

Divide. Write the product as one power.

A.

x10

x9B.

Subtract exponents.x10 – 9

x Think: x = x1

Pre-Algebra

2-7 Properties of Exponents

Subtract exponents.

97

99 – 2

99

92

Try This: Example 2

Divide. Write the product as one power.

A.

B. e10

e5

Subtract exponents.e10 – 5

e5

Pre-Algebra

2-7 Properties of Exponents

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)

Pre-Algebra

2-7 Properties of Exponents

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.

Additional Example 3: Astronomy Application

Pre-Algebra

2-7 Properties of Exponents

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: Example 3

Pre-Algebra

2-7 Properties of Exponents

Lesson Quiz: Part 1

Write the product or quotient as one power

3.

8 9

n 121. (n3)4

109

105 10 4

4.

t 2

5. 33 • 32 • 35 3 10

2. 8 • 88

t9

t7