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## Pre-algebra

### Course: Pre-algebra>Unit 11

Lesson 3: 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...
$\begin{array}{rl}{2}^{2}\cdot {2}^{-2}& ={2}^{2}\cdot \frac{1}{{2}^{2}}\\ \\ & =\frac{{2}^{2}}{{2}^{2}}\\ \\ & =1\end{array}$