If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Algebra basics (ASL)

### Course: Algebra basics (ASL)>Unit 6

Lesson 2: Negative exponents (ASL)

# 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, start superscript, minus, n, end superscript, equals, start fraction, 1, divided by, x, start superscript, n, end superscript, end fraction

### Examples

• 3, start superscript, minus, 5, end superscript, equals, start fraction, 1, divided by, 3, start superscript, 5, end superscript, end fraction
• start fraction, 1, divided by, 2, start superscript, 8, end superscript, end fraction, equals, 2, start superscript, minus, 8, end superscript
• y, start superscript, minus, 2, end superscript, equals, start fraction, 1, divided by, y, squared, end fraction
• left parenthesis, start fraction, 8, divided by, 6, end fraction, right parenthesis, start superscript, minus, 3, end superscript, equals, left parenthesis, start fraction, 6, divided by, 8, end fraction, right parenthesis, cubed

### Practice

Problem 1
• Current
Select the equivalent expression.
4, start superscript, minus, 3, end superscript, equals, question mark

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

n2, start superscript, n, end superscript
32, cubed, equals, 8
22, squared, equals, 4
12, start superscript, 1, end superscript, equals, 2
02, start superscript, 0, end superscript, equals, 1
minus, 12, start superscript, minus, 1, end superscript, equals, start fraction, 1, divided by, 2, end fraction
minus, 22, start superscript, minus, 2, end superscript, equals, start fraction, 1, divided by, 4, end fraction
Notice how 2, start superscript, n, end superscript 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 start fraction, x, start superscript, n, end superscript, divided by, x, start superscript, m, end superscript, end fraction, equals, x, start superscript, n, minus, m, end superscript. 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, start superscript, minus, 1, end superscript, equals, start fraction, 1, divided by, 2, end fraction.
Also, recall that x, start superscript, n, end superscript, dot, x, start superscript, m, end superscript, equals, x, start superscript, n, plus, m, end superscript. 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}