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

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$\displaystyle 2$

   Steps

   1. $\displaystyle 8^{\frac{1}{3}}$$\displaystyle = (2^{3})^{\frac{1}{3}}$$\displaystyle = 2$
2. 02What is the value?$\displaystyle a^{0} \quad (a \neq 0)$AnswerHide

   Answer$\displaystyle 1$

   Steps

   1. $\displaystyle a^{3} \div a^{3}$$\displaystyle = 1$
   2. Write the same thing two ways
   3. $\displaystyle a^{3} \div a^{3}$$\displaystyle = a^{3-3} = a^{0}$
3. 03Write as a fraction:$\displaystyle 2^{-3}$AnswerHide

   Answer$\displaystyle \frac{1}{8}$

   Steps

   1. $\displaystyle 2^{-3}$$\displaystyle = \frac{1}{2^{3}}$$\displaystyle = \frac{1}{8}$
4. 04What number goes in □?$\displaystyle a^{5} \div a^{2} = a^{\square}$AnswerHide

   Answer$\displaystyle 3$

   Steps

   1. $\displaystyle a^{5} \div a^{2}$$\displaystyle = a^{5-2}$$\displaystyle = a^{3}$
5. 05Evaluate:$\displaystyle (2^{3})^{3}$AnswerHide

   Answer$\displaystyle 2^{9}$

   Steps

   1. $\displaystyle (2^{3})^{3}$$\displaystyle = 2^{3 \times 3}$$\displaystyle = 2^{9}$
6. 06Write as a fraction:$\displaystyle 5^{-2}$AnswerHide

   Answer$\displaystyle \frac{1}{25}$

   Steps

   1. $\displaystyle 5^{-2}$$\displaystyle = \frac{1}{5^{2}}$$\displaystyle = \frac{1}{25}$

FAQ