• To investigate the meaning of non-positive exponents.• 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.

• To write in scientific notation numbers close to zero.

Have you noticed that so far in this chapter the exponents have been positive integers?

In this lesson you will learn what a zero or a negative integer means as an exponent.

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

xx

8zz

5

22

3

3

22

5

5

xx

6

3

mm

3

5

55

2y 23 07 42 3x

7z 02 0x 3m 25

Some of your answers in Step 1 should have been positive exponents, some should have been negative, and some should have been

zero exponents.

•Step 2▫How can you tell what type of exponent will

result simply by looking at the original expression?

7

5

yy

2

4

33

4

4

77

3

6

xx

8zz

5

22

3

3

22

5

5

xx

6

3

mm

3

5

55

2y 23 07 42 3x

7z 02 0x 3m 25

•Step 3▫Go back to the expressions in Step 1 that

resulted in a negative exponent. Write each in expanded form. Then reduce them.

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 4▫Compare your answer from step 3 and Step

1. Tell what a base raised to a negative exponent means.

•Step 5▫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 6▫Compare your answers from Step 5 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 A

•Use the properties of exponents to rewrite each expression without a fraction bar.

5

7

34 8

25x

3

8

52

8

8

3(17)17

55 5 7

7 7

3 13 3 4

4 4 8

8

125 25 x

x

3 3 88

15 5 2

2

03 17 3

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 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.

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