# How do you simplify (sqrtx+sqrty)/(sqrtx-sqrty)?

Feb 9, 2017

The answer is $= \frac{x + y + 2 \sqrt{x y}}{x - y}$

#### Explanation:

Multiply numerator and denominator by $\sqrt{x} + \sqrt{y}$

So,

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

$= \frac{\sqrt{x} + \sqrt{y}}{\sqrt{x} - \sqrt{y}} \cdot \frac{\sqrt{x} + \sqrt{y}}{\sqrt{x} + \sqrt{y}}$

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

Feb 9, 2017

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

#### Explanation:

To "rationalise" the denominator, multiply by its radical conjugate $\sqrt{x} + \sqrt{y}$ as follows:

(sqrt(x)+sqrt(y))/(sqrt(x)-sqrt(y)) = ((sqrt(x)+sqrt(y))(sqrt(x)+sqrt(y)))/((sqrt(x)-sqrt(y))(sqrt(x)+sqrt(y))

$\textcolor{w h i t e}{\frac{\sqrt{x} + \sqrt{y}}{\sqrt{x} - \sqrt{y}}} = \frac{x + 2 \sqrt{x} \sqrt{y} + y}{x - y}$

$\textcolor{w h i t e}{\frac{\sqrt{x} + \sqrt{y}}{\sqrt{x} - \sqrt{y}}} = \frac{x + 2 \sqrt{x y} + y}{x - y}$