# How do you write sqrt(7mn) ^5 as an exponential form?

Jun 14, 2017

${\left(7 m n\right)}^{\frac{5}{2}}$

#### Explanation:

Use the law of indices which states:

${\left(\sqrt[q]{x}\right)}^{p} = {x}^{\frac{p}{q}}$

The root becomes the denominator of the fraction in the index, and the power becomes the numerator in the fraction:

$\sqrt[3]{{x}^{4}} = {x}^{\frac{4}{3}}$

In this case we have:

${\left(\sqrt{7 m n}\right)}^{5} = {\left(7 m n\right)}^{\frac{5}{2}}$

Jun 14, 2017

sqrt(7mn)^5=color(red)((7mn)^(5/2)

#### Explanation:

${\sqrt{7 m n}}^{5} = {\left(\sqrt{7 m n}\right)}^{5}$ (at least that's what I assumed was meant by the question)

$\textcolor{w h i t e}{\text{XXX}} = {\left({\left(7 m n\right)}^{\frac{1}{2}}\right)}^{5}$

$\textcolor{w h i t e}{\text{XXX}} = {\left(7 m n\right)}^{\frac{5}{2}}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

If this were intended to be the 5th root, it should have been written as:
$\textcolor{w h i t e}{\text{XXX}} \sqrt[5]{7 m n}$

If the 5 were to apply only to the $n$ then it should appear under the radical sign as
$\textcolor{w h i t e}{\text{XXX}} \sqrt{7 m {n}^{5}}$