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### Course: 8th grade > Unit 1

Lesson 7: Negative exponents# Negative exponents review

Review the basics of negative exponents and try some practice problems.

## Definition for negative exponents

We define a negative power as the multiplicative inverse of the base raised to the positive opposite of the power:

*Want to learn more about this definition? Check out this video.*

### Examples

${3}^{-5}={\displaystyle \frac{1}{{3}^{5}}}$ $\frac{1}{{2}^{8}}}={2}^{-8$ ${y}^{-2}={\displaystyle \frac{1}{{y}^{2}}}$ ${\left({\displaystyle \frac{8}{6}}\right)}^{-3}={\left({\displaystyle \frac{6}{8}}\right)}^{3}$

## Some intuition

So why do we define negative exponents this way? Here are a couple of justifications:

### Justification #1: Patterns

Notice how ${2}^{n}$ is divided by $2$ each time we reduce $n$ . This pattern continues even when $n$ is zero or negative.

### Justification #2: Exponent properties

Recall that $\frac{{x}^{n}}{{x}^{m}}}={x}^{n-m$ . So...

We also know that

And so we get ${2}^{-1}={\displaystyle \frac{1}{2}}$ .

Also, recall that ${x}^{n}\cdot {x}^{m}={x}^{n+m}$ . So...

And indeed, according to the definition...

## Want to join the conversation?

- how can you say that 1/1/9 is 9??(13 votes)
- 1/(1/9). How many 9ths are in one whole? Nine.(67 votes)

- how do we divide exponents by exponents?(8 votes)
- you subtract the exponent on the top from the exponent on the bottom.(28 votes)

- What happens when zero is put to the zero power, for example 0^0(5 votes)
- Interesting question! Consider the following two rules:

1) Any nonzero number to the zero power is 1.

2) Zero to any positive power is 0.

If we try to extend both rules to define 0^0, we get different answers. So should 0^0 be 0, 1, or something else? Because of this situation, it is best to call 0^0 indeterminate (though 0^0 is often interpreted as 1).

Have a blessed, wonderful day!(28 votes)

- wowzers i really had a blast(13 votes)
- how much math is too much math?(9 votes)
- any math. all math is to much math(6 votes)

- Me too but I guess we just have to learn it(2 votes)

- if a exponent is negative what happens to the base(4 votes)
- The base remains the same. As the page explains, a negative exponent just means "the multiplicative inverse of the base raised to the positive opposite of the power". So a^(-b) = 1/(a^b). The base, a, doesn't change. Only its place in the expression changes.(9 votes)

- Man, I thought I was going lose!(7 votes)
- This was so helpful(6 votes)
- i need more practice(6 votes)