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# Fractional indices

A positive number can be raised to a fractional index.

$27^{1/3}$, $100^{3/2}$ and $\pi^{-5/7}$ have fractional indices.

For a positive number $a$, $a^{1/\Torange{q}}$ is the positive number that, when raised to the power of $\Torange{q}$, is equal to a. It is called the $q$-th root of $a$. It is also written as $\sqrt[\Torange{q}]{a}$.

$$\Tblue{9}^{1/\Torange{2}} = \sqrt{\Tblue{9}} = 3,\quad 3^\Torange{2} = \Tblue{9},\qquad \Tblue{16}^{1/\Torange{4}} = \sqrt[\Torange{4}]{\Tblue{16}} = 2,\quad 2^\Torange{4} =\Tblue{16}.$$

$a^{1/2}$ is the square root $\sqrt{a}$ of $a$. $a^{1/3}$ is the cube root.

For a positive number $a$, $a^{\Tred{p}/\Torange{q}}$ is $a^{1/\Torange{q}}$ raised to the power of $\Tred{p}$.

\begin{align*} &\qquad \Tblue{27}^{\Tred{2}/\Torange{3}} = (\Tblue{27}^{1/\Torange{3}})^\Tred{2} = 3^\Tred{2} = 9,\\ &\Tblue{100}^{\Tred{-3}/\Torange{2}} = (\Tblue{100}^{1/\Torange{2}})^{\Tred{-3}} = 10^{\Tred{-3}} = 0.001. \end{align*}