# How do you simplify square root of three to the 15th power?

## $\sqrt{\left({3}^{15}\right)}$

Jan 27, 2018

$\sqrt{{3}^{15}} = {\left(\sqrt{3}\right)}^{15} = 2187 \sqrt{3}$

#### Explanation:

The question is slightly ambiguous in that it could mean either of the following:

• Take the square root of $3$ then raise it to the $15$th power, i.e. ${\left(\sqrt{3}\right)}^{15}$

• Raise $3$ to the $15$th power then take the square root, i.e. $\sqrt{{3}^{15}}$

In general if $a \ge 0$ then $\sqrt{{a}^{2} b} = a \sqrt{b}$

So we find:

$\sqrt{{3}^{15}} = \sqrt{{\left({3}^{7}\right)}^{2} \cdot 3} = {3}^{7} \sqrt{3} = 2187 \sqrt{3}$

Also:

${\left(\sqrt{3}\right)}^{15} = {\left(\sqrt{3}\right)}^{14} \sqrt{3} = {\left({\left(\sqrt{3}\right)}^{2}\right)}^{7} \cdot \sqrt{3} = {3}^{7} \sqrt{3} = 2187 \sqrt{3}$