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What Is the Connection Between Powers and Roots?

a^{b}

consists of a base $a$ and an exponent $b$ . The base $a$ is the number that is to be multiplied with itself $b$ times.

\begin{matrix} a^{b} = \underset{b times}{\underset{⏟}{a \cdot a \cdot a \dots a}} \\ \underset{b times}{\underset{⏟}{\sqrt[b]{a} \cdot \sqrt[b]{a} \cdot \sqrt[b]{a} \dots \sqrt[b]{a}}} = {(\sqrt[b]{a})}^{b} = a \end{matrix}

Rule

The Power Rules

\begin{array}{l} a^{0} & = 1 \\ {(a^{p})}^{q} & = a^{p \cdot q} \\ \frac{1}{a^{- q}} & = a^{q} \\ a^{\frac{p}{q}} & = \sqrt[q]{(a^{p})} \\ a^{p} \cdot a^{q} & = a^{p + q} \\ a^{\frac{p}{q}} & = {(\sqrt[q]{a})}^{p} \\ \frac{a^{p}}{a^{q}} & = a^{p - q} \\ \sqrt[q]{a} \sqrt[q]{b} & = \sqrt[q]{a b} \\ {(a \cdot b)}^{p} & = a^{p} \cdot b^{p} \\ \frac{\sqrt[q]{a}}{\sqrt[q]{b}} & = \sqrt[q]{\frac{a}{b}} \\ {(\frac{a}{b})}^{p} & = \frac{a^{p}}{b^{p}} \end{array}

\begin{array}{l} a^{0} & = 1 & {(a^{p})}^{q} & = a^{p \cdot q} \\ \frac{1}{a^{- q}} & = a^{q} & a^{\frac{p}{q}} & = \sqrt[q]{(a^{p})} \\ a^{p} \cdot a^{q} & = a^{p + q} & a^{\frac{p}{q}} & = {(\sqrt[q]{a})}^{p} \\ \frac{a^{p}}{a^{q}} & = a^{p - q} & \sqrt[q]{a} \sqrt[q]{b} & = \sqrt[q]{a b} \\ {(a \cdot b)}^{p} & = a^{p} \cdot b^{p} & \frac{\sqrt[q]{a}}{\sqrt[q]{b}} & = \sqrt[q]{\frac{a}{b}} \\ {(\frac{a}{b})}^{p} & = \frac{a^{p}}{b^{p}} \end{array}

Note! $\frac{1}{a} = a^{- 1}$

Pay special attention to the fact that $\sqrt{a} = a^{\frac{1}{2}}$ . This is called the square root of $a$ . It then follows that $\sqrt[n]{a} = a^{\frac{1}{n}}$ . This is called the $n$ th root of $a$ . If $n$ is an even number, you get an even root. Some examples include $\sqrt[4]{a}, \sqrt[6]{a}, \sqrt[8]{a}, \dots$

In the same way, you get odd roots when $n$ is an odd number. They look like this: $\sqrt[3]{a}, \sqrt[5]{a}, \sqrt[7]{a}, \dots$

The difference between even and odd roots is that even roots are only defined for $a \geq 0$ . In odd roots, $a$ can be either positive or negative.

Note! You can not take the even root of a negative number. $\sqrt[4]{- 2} =$ is undefined!

Example 1

Write $\frac{a^{0} 2^{3} b}{a^{2} b^{- 2}}$ as simply as possible

\frac{a^{0} 2^{3} b}{a^{2} b^{- 2}} = \frac{1 \cdot 8 \cdot b^{1 + 2}}{a^{2}} = \frac{8 b^{3}}{a^{2}}

Example 2

Write $\frac{3^{2} \cdot x y^{- 4}}{x^{- 4} y^{- 2}}$ as simply as possible

\frac{3^{2} \cdot x y^{- 4}}{x^{- 4} y^{- 2}} = \frac{9 x^{1 + 4}}{y^{- 2 + 4}} = \frac{9 x^{5}}{y^{2}}

Example 3

Write $\frac{2 a^{4} \cdot {(a b)}^{- 4} \cdot b^{5}}{{(a^{2} b)}^{- 6} \cdot 2 b^{5}}$ as simply as possible

\begin{array}{l} \frac{2 a^{4} \cdot {(a b)}^{- 4} \cdot b^{5}}{{(a^{2} b)}^{- 6} \cdot 2 b^{5}} & = \frac{2 a^{4} a^{- 4} b^{- 4} b^{5}}{a^{- 12} b^{- 6} 2 b^{5}} \\ = 2^{1 - 1} a^{4 - 4 + 12} b^{- 4 + 6 + 5 - 5} \\ = 2^{0} a^{12} b^{2} \\ = a^{12} b^{2} \end{array}

\begin{array}{l} \frac{2 a^{4} \cdot {(a b)}^{- 4} \cdot b^{5}}{{(a^{2} b)}^{- 6} \cdot 2 b^{5}} & = \frac{2 a^{4} a^{- 4} b^{- 4} b^{5}}{a^{- 12} b^{- 6} 2 b^{5}} \\ = 2^{1 - 1} a^{4 - 4 + 12} b^{- 4 + 6 + 5 - 5} \\ = 2^{0} a^{12} b^{2} \\ = a^{12} b^{2} \end{array}

Example 4

Write $\frac{2 a^{\frac{3}{2}} b}{2 a^{- \frac{3}{2}}}$ as simply as possible

\frac{2 a^{\frac{3}{2}} b}{2 a^{- \frac{3}{2}}} = \frac{2 a^{\frac{3}{2} + \frac{3}{2}} b}{2} = a^{\frac{6}{2}} b = a^{3} b

Note! When you have a negative exponent, it’s usually best to write the power as a fraction. The reason to do this is that in general, positive exponents are much easier to deal with than negative ones.

Examples of Solving Powers with Variables

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