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

Jan 31, 2016

The square root is expressed as an exponent of $\frac{1}{2}$, so $\sqrt{{x}^{5}}$ can be expressed as ${x}^{\frac{5}{2}}$.

#### Explanation:

Roots are expressed as fractional exponents:

$\sqrt[2]{x} = {x}^{\frac{1}{2}}$
$\sqrt[3]{x} = {x}^{\frac{1}{3}}$

and so on.

This makes sense, because when we multiply we add exponents:

$\sqrt{x}$ x $\sqrt{x}$ = $x$

${x}^{\frac{1}{2}}$ x ${x}^{\frac{1}{2}}$ = ${x}^{\left(\frac{1}{2} + \frac{1}{2}\right)}$ = ${x}^{1}$ = $x$

When an exponent is raised to another exponent, the exponents are multiplied:

$\sqrt{{x}^{5}} = {\left({x}^{5}\right)}^{\frac{1}{2}} = {x}^{5 \cdot \frac{1}{2}} = {x}^{\frac{5}{2}}$