# How do you rationalize the denominator and simplify (sqrtx + 2sqrty)/(sqrtx - 2sqrty)?

Apr 3, 2016

$\frac{x + 4 \sqrt{x y} + 4 y}{x - 4 y}$

#### Explanation:

Multiply the fraction by the conjugate of its denominator.

$= \frac{\sqrt{x} + 2 \sqrt{y}}{\sqrt{x} - 2 \sqrt{y}} \left(\frac{\sqrt{x} + 2 \sqrt{y}}{\sqrt{x} + 2 \sqrt{y}}\right)$

In the denominator, we have what will become a difference of squares, which take the form:

$\left(a + b\right) \left(a - b\right) = {a}^{2} - {b}^{2}$

Here, we have $a = \sqrt{x}$, so ${a}^{2} = x$, and $b = 2 \sqrt{y}$, which implies that ${b}^{2} = 4 y$.

Thus, the fraction simplifies to be

$= {\left(\sqrt{x} + 2 \sqrt{y}\right)}^{2} / \left(x - 4 y\right)$

This is a simplified answer. However, another perfectly acceptable way of presenting this would be to distribute the squared binomial as follows:

$= \frac{\left(\sqrt{x} + 2 \sqrt{y}\right) \left(\sqrt{x} + 2 \sqrt{y}\right)}{x - 4 y} = \frac{x + 4 \sqrt{x y} + 4 y}{x - 4 y}$