All Power Rules

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

Rule

\begin{array}{l} a^{n} \cdot a^{m} & = a^{n + m} \\ \frac{a^{n}}{a^{m}} & = a^{n - m} \\ {(a^{b})}^{c} & = a^{b \cdot c} \\ {(a \cdot b)}^{n} & = a^{n} \cdot b^{n} \\ {(\frac{a}{b})}^{n} & = \frac{a^{n}}{b^{n}} \\ \sqrt{a} & = a^{\frac{1}{2}} \\ {(\sqrt[n]{a})}^{m} & = \sqrt[n]{a^{m}} = a^{\frac{m}{n}} \\ {(\sqrt[n]{a \cdot b})}^{m} & = \sqrt[n]{a^{m}} \cdot \sqrt[n]{b^{m}} \\ = a^{\frac{m}{n}} \cdot b^{\frac{m}{n}} \\ a^{- n} & = \frac{1}{a^{n}} \\ a^{0} & = 1 \\ a^{1} & = a \end{array}

\begin{array}{l} a^{n} \cdot a^{m} & = a^{n + m} \\ \frac{a^{n}}{a^{m}} & = a^{n - m} \\ {(a^{b})}^{c} & = a^{b \cdot c} \\ {(a \cdot b)}^{n} & = a^{n} \cdot b^{n} \\ {(\frac{a}{b})}^{n} & = \frac{a^{n}}{b^{n}} \\ \sqrt{a} & = a^{\frac{1}{2}} \\ {(\sqrt[n]{a})}^{m} & = \sqrt[n]{a^{m}} = a^{\frac{m}{n}} \\ {(\sqrt[n]{a \cdot b})}^{m} & = \sqrt[n]{a^{m}} \cdot \sqrt[n]{b^{m}} = a^{\frac{m}{n}} \cdot b^{\frac{m}{n}} \\ a^{- n} & = \frac{1}{a^{n}} \\ a^{0} & = 1 \\ a^{1} & = a \end{array}

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

Example 1

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

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

\begin{array}{l} 2^{3} \cdot a^{3} \cdot a^{2} \cdot b^{2} \\ = 8 \cdot a^{3} \cdot a^{2} \cdot b^{2} \\ = 8 \cdot a^{3 + 2} \cdot b^{2} \\ = 8 a^{5} b^{2} \end{array}

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