# What is the square root of 156.25?

Sep 18, 2015

$\sqrt{156.25} = \frac{25}{2}$

#### Explanation:

$15625 = {5}^{6}$

So $156.25 = \frac{{5}^{6}}{100} = \frac{{5}^{6}}{{2}^{2} \cdot {5}^{2}} = {5}^{4} / {2}^{2}$

If $a , b \ge 0$, then $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

If $a , b , c > 0$ then ${a}^{b c} = {\left({a}^{b}\right)}^{c}$

So $\sqrt{156.25} = \sqrt{{5}^{4} / {2}^{2}} = \frac{\sqrt{{5}^{4}}}{\sqrt{{2}^{2}}} = \frac{\sqrt{{\left({5}^{2}\right)}^{2}}}{\sqrt{{2}^{2}}} = {5}^{2} / 2 = \frac{25}{2}$