• OUR GOAL: TO WRITE A NUMBER WITH A NEGATIVE EXPONENT IN A

FORM THAT HAS A POSITIVE EXPONENT AND WRITE A NUMBER WITH A POSITIVE EXPONENT IN A

FORM THAT HAS A NEGATIVE EXPONENT.

Lesson 4.6: Zero and Negative Exponents

In this lesson you will learn what a zero or a negative integer means as an exponent. You will then learn how to manipulate them so you always have positive exponents as your answer.

0x 2x

More Exponents

Step 1 Use the division property of exponents to rewrite

each of these expression with a single exponent.

7

5

yy

2

4

33

4

4

77

3

6

xx5

22

2y 23 07 42 3x

Notice that some of your answers in Step 1 are positive exponents, some are negative, and some are

zero exponents.

Step 2 Go back to the expressions in Step 1 that resulted in

a negative exponent. Write each in expanded form (using the cross out method). Then write the answer as a fraction.

2

4

33

3

6

xx5

22

3

5

55

23 42 3x 25

2

3 33 3 3 3

1 193

4

22 2 2 2 2

1 1162

3

1

x x xx x x x x x

x

2

5 5 55 5 5 5 5

1 1255

Step 3 Compare your answer from step 2 and Step 1. Tell

what a base raised to a negative exponent means.

Step 4 Go back to the expressions in Step 1 that resulted in

an exponent of zero. Write each in expanded form. Then reduce them.

4

4

77

3

3

22

5

5

xx

2 2 21

2 2 2

7 7 7 71

7 7 7 7

1x x x x xx x x x x

Step 5 Compare your answers from Step 4 and Step 1. Tell

what a base raised to an exponent of zero means.

Step 7 Use what you have learned about negative exponents

to rewrite each of these expressions with positive exponents and only one fraction bar.

251

8

13

2

2 5

4xz y

2

15

8

13

5

2 2

4yx z

Step 8 In one or two sentences, explain how to rewrite a

fraction with a negative exponent in the numerator or denominator as a fraction with positive exponents.

Exponential Form Fraction Form

33 27

32 9

31 3

30 1

3-1 1/3

3-2 1/9

3-3 1/27

3

For any nonzero value of b and for any value of n,

1nn

bb

1 nn

bb

0 1b

Example B

Solomon bought a used car for $5,600. He estimates that it has been decreasing in value by 15% each year. If his estimate of the rate of depreciation is correct,

how much was the car worth 3 years ago?

If the car is 7 years old, what was the original price of the car?

3(1 ) 5600(1 0.15) $9,118.66xy A r

75600(1 0.15) $17, 468.50

Example CConvert each number to standard notation

from scientific notation, or vice versa. A pi meson, an unstable particle released in a

nuclear reaction, “lives” only 0.000000026 seconds. The number 6.67 x 10-11 is the gravitational constant

in the metric system used to calculate the gravitational attraction between two objects that have given masses and are a given distance apart.

The mass of an electron is 9.1 x 10-31 kg.

Example C Convert each number to standard notation from scientific

notation, or vice versa. A pi meson, an unstable particle released in a nuclear reaction,

“lives” only 0.000000026 seconds.

The number 6.67 x 10-11 is the gravitational constant in the metric system used to calculate the gravitational attraction between two objects that have given masses and are a given distance apart.

The mass of an electron is 9.1 x 10-31 kg.

88

2.6 2.60.000000026 2.6 x10

100,000,000 10

1111

6.676.67 10 0.0000000000667

10x

3131

9.19.1 10 0.00000000000000000000000000000091

10x