# How do you rationalize the denominator and simplify sqrt((20y)/(5x))?

##### 1 Answer
Mar 31, 2016

$\sqrt{\frac{20 y}{5 x}} = \frac{2 \sqrt{x y}}{\left\mid x \right\mid}$

#### Explanation:

We begin by noting that 20 is divisible by 5. Hence,

$\sqrt{\frac{20 y}{5 x}} = \sqrt{\frac{4 y}{x}}$

From the law of indices, we know that

$\sqrt{\frac{a b}{c}} = \sqrt{a} \sqrt{\frac{b}{c}}$ if $a \ge 0$

So,

$\sqrt{\frac{4 y}{x}} = \sqrt{4} \sqrt{\frac{y}{x}}$

$= 2 \sqrt{\frac{y}{x}}$

To rationalize the denominator, we multiply $x$ on both the numerator and the denominator.

$2 \sqrt{\frac{y}{x}} = 2 \sqrt{\frac{y \times x}{x \times x}}$

$= 2 \sqrt{\frac{x y}{{x}^{2}}}$

And from the fact $\sqrt{{x}^{2}} = \left\mid x \right\mid$,

$2 \sqrt{\frac{x y}{{x}^{2}}} = \frac{2 \sqrt{x y}}{\left\mid x \right\mid}$