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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*}