Properties of Properties of ExponentsExponents

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

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.

D. 244 • 244

C. 25 • 2

26

25 + 1

248

24 4 + 4

Think: 2 = 2 1

Multiplying Powers with the Same Base Continued

Multiply. Write the product as one power.

Add exponents.

Add 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

Subtract exponents.

72

75 – 3

75

73

Dividing Powers with the Same Base

Divide. Write the quotient as one power.

A.

x10

x9B.

Subtract exponents.x10 – 9

x Think: x = x1

Subtract exponents.

97

99 – 2

99

92

Try This:

Divide. Write the quotient as one power.

A.

B. e10

e5

Subtract exponents.e10 – 5

e5

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! It is undefined!

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

Practice

Write the product or quotient as one power

1. n3 n4

2.

3.

4. 33 • 32 • 35

5. 8 • 88 =

109

105

t9

t7

1. n7

2. 104

3. t2

4. 310

5. 89