# 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:
$x^{-n}=\dfrac{1}{x^n}$

### Examples

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

### Practice

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}=\dfrac12$
$-2$$2^{-2}=\dfrac14$
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 $\dfrac{x^n}{x^m}=x^{n-m}$. So...
\begin{aligned} \dfrac{2^2}{2^3}&=2^{2-3} \\\\ &=2^{-1} \end{aligned}
We also know that
\begin{aligned} \dfrac{2^2}{2^3}&=\dfrac{\cancel 2\cdot\cancel 2}{\cancel 2\cdot\cancel 2\cdot 2} \\\\ &=\dfrac12 \end{aligned}
And so we get $2^{-1}=\dfrac12$.
Also, recall that $x^n\cdot x^m=x^{n+m}$. So...
\begin{aligned} 2^2\cdot 2^{-2}&=2^{2+(-2)} \\\\ &=2^0 \\\\ &=1 \end{aligned}
And indeed, according to the definition...
\begin{aligned} 2^2\cdot 2^{-2}&=2^2\cdot\dfrac{1}{2^2} \\\\ &=\dfrac{2^2}{2^2} \\\\ &=1 \end{aligned}