# How do you simplify sqrt(196y^10)?

Sep 10, 2015

$\sqrt{196 {y}^{10}} = 14 {y}^{5}$

#### Explanation:

$14 \times 14 = 196$
${y}^{5} \times {y}^{5} = {y}^{10}$

So $\sqrt{196 {y}^{10}} = \sqrt{{14}^{2} {\left({y}^{5}\right)}^{2}}$

$\textcolor{w h i t e}{\text{XXXXXX}} = \left\mid 14 {y}^{5} \right\mid$
...we take the absolute value to ensure the root extracted is the principal root

Sep 10, 2015

$\sqrt{196 {y}^{10}} = 14 \left\mid {y}^{5} \right\mid$

#### Explanation:

First, note that in general $\sqrt{{x}^{2}} = \left\mid x \right\mid$ rather than $x$,

since:

$\sqrt{{x}^{2}} = \left\{\begin{matrix}x & \text{if x >= 0" \\ -x & "if x < 0}\end{matrix}\right.$

Also, if $a , b \ge 0$ then $\sqrt{a b} = \sqrt{a} \sqrt{b}$

So:

$\sqrt{196 {y}^{10}} = \sqrt{{14}^{2} {\left({y}^{5}\right)}^{2}} = \sqrt{{14}^{2}} \sqrt{{\left({y}^{5}\right)}^{2}} = 14 \left\mid {y}^{5} \right\mid$