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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:
${x}^{-n}=\frac{1}{{x}^{n}}$

### Examples

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

### Practice

Problem 1
Select the equivalent expression.
${4}^{-3}=?$

Want to try more problems like these? Check out this exercise.

## Some intuition

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

### Justification #1: Patterns

$n$${2}^{n}$
$3$${2}^{3}=8$
$2$${2}^{2}=4$
$1$${2}^{1}=2$
$0$${2}^{0}=1$
$-1$${2}^{-1}=\frac{1}{2}$
$-2$${2}^{-2}=\frac{1}{4}$
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...
$\begin{array}{rl}\frac{{2}^{2}}{{2}^{3}}& ={2}^{2-3}\\ \\ & ={2}^{-1}\end{array}$
We also know that
$\begin{array}{rl}\frac{{2}^{2}}{{2}^{3}}& =\frac{\overline{)2}\cdot \overline{)2}}{\overline{)2}\cdot \overline{)2}\cdot 2}\\ \\ & =\frac{1}{2}\end{array}$
And so we get ${2}^{-1}=\frac{1}{2}$.
Also, recall that ${x}^{n}\cdot {x}^{m}={x}^{n+m}$. So...
$\begin{array}{rl}{2}^{2}\cdot {2}^{-2}& ={2}^{2+\left(-2\right)}\\ \\ & ={2}^{0}\\ \\ & =1\end{array}$
And indeed, according to the definition...
$\begin{array}{rl}{2}^{2}\cdot {2}^{-2}& ={2}^{2}\cdot \frac{1}{{2}^{2}}\\ \\ & =\frac{{2}^{2}}{{2}^{2}}\\ \\ & =1\end{array}$

## Want to join the conversation?

• how can you say that 1/1/9 is 9??
• 1/(1/9). How many 9ths are in one whole? Nine.
• how do we divide exponents by exponents?
• you subtract the exponent on the top from the exponent on the bottom.
• What happens when zero is put to the zero power, for example 0^0
• 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!
• how much math is too much math?
• any math. all math is to much math
• wowzers i really had a blast
• if a exponent is negative what happens to the base
• 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.
• what do you do if the question is not 2 x 2 and its something like 2 x 3 then what number to you put down because if its the same number you can just get rid of one of them but if there not what do you do
• Look at the hints. The hints are very helpful for exponents.
(1 vote)
• I do not understand why it becomes a fraction. The intuition does not help either. I know it becomes a fraction, i know the right answer i just do not understand it. For example: 2^-4 i do not get why it becomes 1/2^4