# Question #df28f

##### 1 Answer
Oct 5, 2017

${\left(\sqrt{5 a b}\right)}^{3}$

#### Explanation:

Let's begin with something easy. You mentioned that you knew how to convert something such as ${3}^{\frac{1}{4}}$ which resulted in $\sqrt[4]{3}$.

Looking at ${\left(5 a b\right)}^{\frac{3}{2}}$. In radical form this is ${\left(\sqrt{5 a b}\right)}^{3}$.

How?

Well there is a technique to this.

If we apply this to the first example: ${3}^{\frac{1}{4}}$ we can take note that
Power $= \textcolor{red}{1}$
Index$= \textcolor{b l u e}{4}$

(I color coded to better represent the power and index)

So
${3}^{\frac{\textcolor{red}{1}}{\textcolor{b l u e}{4}}} = {\left(\sqrt[\textcolor{b l u e}{4}]{3}\right)}^{\textcolor{red}{1}}$

Which is simplifies to $\sqrt[4]{3}$

Going back to ${\left(5 a b\right)}^{\frac{3}{2}}$, we apply the technique to get
Power$= \textcolor{red}{3}$
Index$= \textcolor{b l u e}{2}$

So converting into radical form

${\left(5 a b\right)}^{\frac{\textcolor{red}{3}}{\textcolor{b l u e}{2}}} = {\left(\sqrt[\textcolor{b l u e}{2}]{5 a b}\right)}^{\textcolor{red}{3}}$

Which we simplify to ${\left(\sqrt{5 a b}\right)}^{3}$ because $\sqrt[2]{5 a b}$ and $\sqrt{5 a b}$ are the same but it is preferred to not include the $2$ because its redundant.

Also, the parenthesis simply indicated that whatever was inside there would go inside the square root. Another example would be ${\left(7 a - 3 b\right)}^{\frac{2}{5}}$. This would mean that $7 a - 3 b$ would be inside the square root once we convert it into radical form which eventually is
${\left(\sqrt[5]{7 a - 3 b}\right)}^{2}$