# How do you simplify sqrt(x^14 y^21) / z^-35?

Dec 31, 2015

${x}^{7} {y}^{\frac{21}{2}} {z}^{35}$

#### Explanation:

According to the exponent rule, ${a}^{-} b = \frac{1}{a} ^ b$ , start by rewriting ${z}^{-} 35$ so that it has a positive exponent.

$\frac{\sqrt{{x}^{14} {y}^{21}}}{z} ^ - 35$

$= \frac{\sqrt{{x}^{14} {y}^{21}}}{\frac{1}{z} ^ 35}$

Divide the numerator by the denominator.

$= \sqrt{{x}^{14} {y}^{21}} \div \frac{1}{z} ^ 35$

Simplify.

$= \sqrt{{x}^{14} {y}^{21}} \cdot {z}^{35} / 1$

$= \sqrt{{x}^{14} {y}^{21}} {z}^{35}$

As a shortcut, since you know that ${z}^{-} 35$ has a negative exponent, to make the term have a positive exponent, you could have also just moved it to the numerator and changed the negative exponent to a positive exponent. For example:

$\frac{\sqrt{{x}^{14} {y}^{21}}}{z} ^ - 35 \Rightarrow \sqrt{{x}^{14} {y}^{21}} {z}^{35}$

Going on, recall that sqrt(color(white)(x) is multiplying an exponent by $\textcolor{red}{\frac{1}{2}}$ (or dividing by $2$).

$= \sqrt{{x}^{14} {y}^{21}} {z}^{35}$

$= \sqrt{{x}^{14}} \sqrt{{y}^{21}} {z}^{35}$

$= {x}^{14 \cdot \textcolor{red}{\frac{1}{2}}} {y}^{21 \cdot \textcolor{red}{\frac{1}{2}}} {z}^{35}$

Simplify.

$= {x}^{7} {y}^{\frac{21}{2}} {z}^{35}$