How do you rationalize the denominator and simplify $\frac{h}{\sqrt{x} - \sqrt{x + h}}$ $h / (sqrtx - sqrt( x+h))$ ?

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How do you rationalize the denominator and simplify $\frac{h}{\sqrt{x} - \sqrt{x + h}}$ $h / (sqrtx - sqrt( x+h))$ ?

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Answer 1 · 2016-04-24T20:09:11

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

Explanation:

The difference of squares identity can be written:

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

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

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Multiply numerator and denominator by $(\sqrt{x} + \sqrt{x + h})$ $(sqrt(x)+sqrt(x+h))$ :

$\frac{h}{\sqrt{x} - \sqrt{x + h}}$ $h/(sqrt(x)-sqrt(x+h))$

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

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

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

$=(color(red)(cancel(color(black)(h)))(sqrt(x)+sqrt(x+h)))/(-color(red)(cancel(color(black)(h))))$

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

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How do you rationalize the denominator and simplify $\frac{h}{\sqrt{x} - \sqrt{x + h}}$ $h / (sqrtx - sqrt( x+h))$ ?

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Explanation:

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How do you rationalize the denominator and simplify $\frac{h}{\sqrt{x} - \sqrt{x + h}}$ $h / (sqrtx - sqrt( x+h))$ ?