# All Power Rules

Here are all the power rules compiled into one place for reference.

Rule

### PowerRules

$\begin{array}{llll}\hfill {a}^{n}\cdot {a}^{m}& ={a}^{n+m}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \frac{{a}^{n}}{{a}^{m}}& ={a}^{n-m}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \phantom{\rule{-0.17em}{0ex}}{\left({a}^{b}\right)}^{c}& ={a}^{b\cdot c}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {\left(a\cdot b\right)}^{n}& ={a}^{n}\cdot {b}^{n}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \phantom{\rule{-0.17em}{0ex}}{\left(\frac{a}{b}\right)}^{n}& =\frac{{a}^{n}}{{b}^{n}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \sqrt{a}& ={a}^{\frac{1}{2}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \phantom{\rule{-0.17em}{0ex}}{\left(\sqrt[n]{a}\right)}^{m}& =\sqrt[n]{{a}^{m}}={a}^{\frac{m}{n}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \phantom{\rule{-0.17em}{0ex}}{\left(\sqrt[n]{a\cdot b}\right)}^{m}& =\sqrt[n]{{a}^{m}}\cdot \sqrt[n]{{b}^{m}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & ={a}^{\frac{m}{n}}\cdot {b}^{\frac{m}{n}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {a}^{-n}& =\frac{1}{{a}^{n}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {a}^{0}& =1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {a}^{1}& =a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill {a}^{n}\cdot {a}^{m}& ={a}^{n+m}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \frac{{a}^{n}}{{a}^{m}}& ={a}^{n-m}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \phantom{\rule{-0.17em}{0ex}}{\left({a}^{b}\right)}^{c}& ={a}^{b\cdot c}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {\left(a\cdot b\right)}^{n}& ={a}^{n}\cdot {b}^{n}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \phantom{\rule{-0.17em}{0ex}}{\left(\frac{a}{b}\right)}^{n}& =\frac{{a}^{n}}{{b}^{n}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \sqrt{a}& ={a}^{\frac{1}{2}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \phantom{\rule{-0.17em}{0ex}}{\left(\sqrt[n]{a}\right)}^{m}& =\sqrt[n]{{a}^{m}}={a}^{\frac{m}{n}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \phantom{\rule{-0.17em}{0ex}}{\left(\sqrt[n]{a\cdot b}\right)}^{m}& =\sqrt[n]{{a}^{m}}\cdot \sqrt[n]{{b}^{m}}={a}^{\frac{m}{n}}\cdot {b}^{\frac{m}{n}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {a}^{-n}& =\frac{1}{{a}^{n}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {a}^{0}& =1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {a}^{1}& =a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Remember that ${\left(a+b\right)}^{n}\ne {a}^{n}+{b}^{n}$. This identity is valid only for multiplication and division, not addition and subtraction.

Example 1

Simplify ${\left(2a\right)}^{3}\cdot {\left(a\cdot b\right)}^{2}$

You use the rules above and get ${\left(2a\right)}^{3}={2}^{3}\cdot {a}^{3}$ and ${\left(a\cdot b\right)}^{2}={a}^{2}\cdot {b}^{2}$. This gives you

$\begin{array}{llll}\hfill & \phantom{=}{2}^{3}\cdot {a}^{3}\cdot {a}^{2}\cdot {b}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =8\cdot {a}^{3}\cdot {a}^{2}\cdot {b}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =8\cdot {a}^{3+2}\cdot {b}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =8{a}^{5}{b}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$