# Exponents Cheat Sheet

## The basics of exponents

Reads as “a to the m power”

$\text{Example: }2^{3} = 8 \quad \text{two to the third power equals eight}$

Go back

## Power of zero (0) rule

$\begin{split} a^{0} &= 1 \\ \\ \text{Example: }3^{0} &= 1 \end{split}$

Go back

## Power of zero (1) rule

$\begin{split} a^{1} &= a \\ \\ \text{Example: }3^{1} &= 3 \end{split}$

Go back

## Negative exponent rule

$\begin{split} a^{-m} &= \frac{1}{a^{m}} \\ \\ \text{Example: }x^{-3} &= \frac{1}{x^{3}} \\ 3^{-2} &= \frac{1}{3^{2}} \end{split}$

Go back

## Fraction to a negative exponent rule

$\begin{split} \left ( \frac{a}{b} \right )^{-n} &= \left ( \frac{b}{a} \right )^{n} = \frac{b^{n}}{a^{n}} \\ \\ \text{Example: }\left ( \frac{x}{y} \right )^{-5} &= \left ( \frac{y}{x} \right )^{5} = \frac{y^{5}}{x^{5}} \\ \left ( \frac{2}{4} \right )^{-5} &= \left ( 0.5 \right )^{-5} = 32 \\ \left ( \frac{4}{2} \right )^{5} &= \left ( 2 \right )^{5} = 32 \\ \frac{4^{5}}{2^{5}} &= \frac{1024}{32} = 32 \end{split}$

Go back

## Product rule

$\begin{split} a^{m} \cdot a^{n} &= a^{m+n} \\ \\ \text{Example: }x^{2} \cdot x^{3} &= x^{2+3} = x^{5} \\ 2^{2} \cdot 2^{3} &= 4 \cdot 8 = 32 \\ 2^{5} &= 32 \end{split}$

Go back

## Quotient rule

$\begin{split} \frac{a^{m}}{a^{n}} &= a^{m-n} \\ \\ \text{Example: }\frac{x^{5}}{x^{3}} &= x^{5-3} = x^{2} \\ \frac{2^{5}}{2^{3}} &= \frac{32}{8} = 4 \\ 2^{2} &= 4 \end{split}$

Go back

## Power of a product rule

$\begin{split} (a \cdot b)^{m} &= a^{m} \cdot b^{m} \\ \\ \text{Example: }(x \cdot y)^{3} &= x^{3} \cdot y^{3} \\ (2 \cdot 4)^{3} &= 8^{3} = 512 \\ 2^{3} = 8 \text{, } 4^{3} &= 64 \text{, } 8 \cdot 64 = 512 \end{split}$

Go back

## Power to a power rule

$\begin{split} \left (a^{m} \right )^{n} &= a^{m \cdot n} \\ \\ \text{Example: }\left (x^{2} \right )^{3} &= x^{2 \cdot 3} = x^{6} \\ \left (2^{2} \right )^{3} &= 2^{6} = 64 \\ \left (2^{2} \right )^{3} &= 4^{3} = 64 \end{split}$

Go back

## Power of a quotient rule

$\begin{split} \left (\frac{a}{b} \right )^{m} &= \frac{a^{m}}{b^{m}} \\ \\ \text{Example: }\left (\frac{x}{y} \right )^{3} &= \frac{x^{3}}{y^{3}} \\ \left (\frac{4}{2} \right )^{3} &= \frac{4^{3}}{2^{3}} = \frac{64}{8} = 8 \\ \left (\frac{4}{2} \right )^{3} &= 2^{3} = 8 \end{split}$

Go back

## Power of a fraction rule

$\begin{split} a^{\frac{m}{n}} &= \left ( a^{\frac{1}{n}} \right )^{m} = \left ( a^{m} \right )^{\frac{1}{n}} \\ \\ \text{Example: } x^{\frac{2}{3}} &= \left ( x^{\frac{1}{3}} \right )^{2} = \left ( x^{2} \right )^{\frac{1}{3}} \\ 9^{\frac{4}{2}} &= \left ( 9^{\frac{1}{2}} \right )^{4} = \left ( 3 \right )^{4} = 81 \\ 9^{\frac{4}{2}} &= \left ( 9^{4} \right )^{\frac{1}{2}} = \left ( 6561 \right )^{\frac{1}{2}} = 81 \end{split}$

Go back

Reference:
https://tutorial.math.lamar.edu/Extras/CheatSheets_Tables.aspx