# How do you rationalize the denominator and simplify h / (sqrtx - sqrt( x+h))?

Apr 24, 2016

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

#### Explanation:

The difference of squares identity can be written:

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

We use this with $a = \sqrt{x}$ and $b = \sqrt{x + h}$ later.

$\textcolor{w h i t e}{}$
Multiply numerator and denominator by $\left(\sqrt{x} + \sqrt{x + h}\right)$:

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

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

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

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

$= \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{h}}} \left(\sqrt{x} + \sqrt{x + h}\right)}{- \textcolor{red}{\cancel{\textcolor{b l a c k}{h}}}}$

$= - \left(\sqrt{x} + \sqrt{x + h}\right)$