Negative Exponents

5

2

x

x

• Recall how we divide powers with the same base …

• Use the same property on a different problem …

2

1

x• This suggests that …

5 2x 3x

0

2

x

x 0 2x 2x

22

1x

x

• We will in fact define that if b is a nonzero real number and n is an integer, then

2

1nbb

• Example 1

Simplify: 5x

By the negative exponent definition given above, we conclude that …

55

1x

x

• Example 2

Note that the exponent only applies to the 5, not to the negative.

Simplify: 25

25 21 5 2

11

5

11

25

1

25

• Example 3

In this case, the exponent applies to the -5.

Simplify: 25

25

2

1

5

1

25

• Example 4

Here the negative exponent only applies to the x.

Simplify: 4 32x y

4 32 x y 34

12 yx

3

4

2y

x

In the future we will leave out the middle step with the thinking, if a variable in the numerator has a negative exponent, send it to the denominator with a positive exponent.

• Example 5

Here the negative exponent applies to everything within the parentheses.

Simplify: 12 36x y

12 36x y

12 3

1

6x y

2 3

1

6x y

• Example 6

Simplify: 3

1

x

In the future we will leave out the middle steps with the thinking, if a variable in the denominator has a negative exponent, send it to the numerator with a positive exponent.

3

11

x 3x

3

11x

3

1

x

• For example, that will not work in the following problem:

• Important note: when we talk about sending to the numerator or denominator, it is assumed we have a product, as in the previous problems.

This problem is more complicated, and will be dealt with at a later time.

2 3x y

z

Here we have a sum in the numerator, and cannot simply send the x to the denominator with a positive exponent.

• Example 7

Simplify:

31

2

This problem can be worked in a variety of ways, but the easiest is to simply take the reciprocal of the fraction in parentheses, and change the exponent to a positive.

This would be the same as sending the fraction to the denominator and then multiplying by the reciprocal.

31

2

32

1

32 8