Exponent rules and exponential functions

Two rules do most of the work: multiplying adds exponents, and a power of a power multiplies them.

Exponents extend beyond positive integers: $a^0=1$ ($a\neq 0$), $a^{-n}=\frac{1}{a^n}$, and $a^{1/n}$ is the $n$-th root. For example, $8^{1/3}=2$ and $2^{-3}=\frac{1}{8}$.

In calculus you use these rules directly when differentiating or integrating $e^x$, and when rewriting $a^x$ as $e^{x\ln a}$.

Examples

1. 01Evaluate:$\displaystyle 8^{\frac{1}{3}}$AnswerHide

   Answer$\displaystyle 2$
2. 02What is the value?$\displaystyle a^{0} \quad (a \neq 0)$AnswerHide

   Answer$\displaystyle 1$
3. 03Write as a fraction:$\displaystyle 2^{-3}$AnswerHide

   Answer$\displaystyle \frac{1}{8}$
4. 04Evaluate:$\displaystyle (2^{3})^{3}$AnswerHide

   Answer$\displaystyle 2^{9}$
5. 05Write as a fraction:$\displaystyle 5^{-2}$AnswerHide

   Answer$\displaystyle \frac{1}{25}$
6. 06Write as a fraction:$\displaystyle 5^{-3}$AnswerHide

   Answer$\displaystyle \frac{1}{125}$

FAQ

Q1Why is a^0 equal to 1?Show answerHide

A

For the rule $a^m a^n=a^{m+n}$ to hold with $n=0$ we need $a^m a^0=a^m$, so $a^0=1$ (for $a\neq 0$). The same reasoning fixes $a^{-n}=1/a^n$.

Practice this pattern in the app, 5 problems a day.

Published 2026-09-04 · Updated 2026-09-05