# How do you simplify sqrt(y^6)?

Mar 27, 2016

=color(green)(y^3

#### Explanation:

sqrt ( y^6

Square root , can also be called as second root, in terms of fraction, second root is a half power (color(blue)(1/2)

So,
sqrt ( y^6) = y^(6 * color(blue)(1/2)

$\sqrt{{y}^{6}} = {y}^{3}$

=color(green)(y^3

Mar 27, 2016

Another way of writing the same thing

${y}^{3}$

#### Explanation:

$\textcolor{b l u e}{\text{Method 1}}$

${y}^{6} \equiv {y}^{2} \times {y}^{2} \times {y}^{2}$

So $\sqrt{{y}^{6}} = \sqrt{{y}^{2} \times {y}^{2} \times {y}^{2}} = y \times y \times y$

Thus $\textcolor{g r e e n}{\sqrt{{y}^{6}} = {y}^{3}}$

But #y^3 cab be negative and this solution is always positive so Allan's recommendation ties this down by stating:

Thus $\textcolor{g r e e n}{\sqrt{{y}^{6}} = | {y}^{3} |}$

For any value $x \to | x |$ is always positive

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Method 2}}$

Suppose we had $\sqrt{x}$ then another way of writing this is$\text{ } {x}^{\frac{1}{2}}$

So $\sqrt{{y}^{6}} = {\left({y}^{6}\right)}^{\frac{1}{2}}$

and ${\left({y}^{6}\right)}^{\frac{1}{2}} \text{ is the same as "y^(6xx1/2)" "=" } {y}^{\frac{6}{2}}$

But $\frac{6}{2} = 3$

So $\textcolor{g r e e n}{{y}^{\frac{6}{3}} \text{ is the same as } {y}^{3}}$