How do you simplify $\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} + \sqrt{b}}$ $(sqrta- sqrtb)/(sqrta+sqrtb)$ ?

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How do you simplify $\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} + \sqrt{b}}$ $(sqrta- sqrtb)/(sqrta+sqrtb)$ ?

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Answer 1 · 2015-05-19T16:33:34

Multiply both numerator and denominator by $\sqrt{a} - \sqrt{b}$ $sqrta - sqrtb$

$\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} + \sqrt{b}} = \frac{{(\sqrt{a} - \sqrt{b})}^{2}}{a - b}$ $(sqrta - sqrtb)/(sqrta + sqrtb)= (sqrta - sqrtb)^2/(a - b)$

Answer 2 · 2015-05-19T16:49:39

In this case, simplifying means rationalizing the denominator, that is getting rid of its square roots.

This can be done by using a well known algebraic identity: $x^{2} - y^{2} = (x + y) (x - y)$ $x^2 - y^2 = (x+ y)(x - y)$ .

In our case, we have to multiply the fraction by $(\sqrt{a} - \sqrt{b})$ $(sqrt a - sqrt b)$ :

$\frac{(\sqrt{a} - \sqrt{b}) (\sqrt{a} - \sqrt{b})}{(\sqrt{a} - \sqrt{b}) (\sqrt{a} + \sqrt{b})}$ $((sqrt a - sqrt b)(sqrt a - sqrt b)) / ((sqrt a - sqrt b)(sqrt a + sqrt b))$ = $\frac{{(\sqrt{a} - \sqrt{b})}^{2}}{{(\sqrt{a})}^{2} - {(\sqrt{b})}^{2}}$ $((sqrt a - sqrt b)^2) / ((sqrt a)^2 - (sqrt b)^2)$ = $\frac{{(\sqrt{a} - \sqrt{b})}^{2}}{a - b}$ $((sqrt a - sqrt b)^2) / (a -b)$

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How do you simplify $\frac{\sqrt{a} - \sqrt{b}}{\sqrt{a} + \sqrt{b}}$ $(sqrta- sqrtb)/(sqrta+sqrtb)$ ?

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