# How do you change the expression 5^(1/2) to radical form?

Jun 29, 2016

${5}^{\frac{1}{2}} = \textcolor{b l u e}{\sqrt{5}}$

#### Explanation:

This is basically a definition:
$\textcolor{w h i t e}{\text{XXX}} {b}^{\frac{1}{a}} = \sqrt[a]{b}$

To see why this is a reasonable definition
consider that in general
$\textcolor{w h i t e}{\text{XXX}} {b}^{2 k} = {b}^{k} \cdot {b}^{k}$

In the case of ${5}^{\frac{1}{2}}$
$\textcolor{w h i t e}{\text{XXX}} 5 = {5}^{1} = \left({5}^{\frac{1}{2}}\right) \cdot \left({5}^{\frac{1}{2}}\right)$
that is
$\textcolor{w h i t e}{\text{XXX}} {5}^{\frac{1}{2}}$ must be a value which when multiplied by itself is equal to $5$
and since $\sqrt{5} \cdot \sqrt{5} = 5$, the primary solution to this is
$\textcolor{w h i t e}{\text{XXX}} {5}^{\frac{1}{2}} = \sqrt{5}$