Question

Rationalise the denominator and simplify
(√5/√6+2)-(√5/√6-2)

Answer 1

The simplified answer to the given expression is $-2\sqrt{5}$.

Explanation:

The given expression : $(\dfrac{\sqrt{5}}{\sqrt{6}+2})-(\dfrac{\sqrt{5}}{\sqrt{6}-2})$

First we rationalize  $\dfrac{\sqrt{5}}{\sqrt{6}+2}$ by multiplying $\sqrt{6}-2$ to the numerator and denominator  , we get

$\dfrac{\sqrt{5}}{\sqrt{6}+2}\times\dfrac{\sqrt{6}-2}{\sqrt{6}-2}\\\\\\=\dfrac{\sqrt{5}(\sqrt{6}-2)}{(\sqrt{6})^2-2^2}\ \ [\because\ (a+b)(a-b)=a^2-b^2]\\\\\\=\dfrac{\sqrt{5}(\sqrt{6})-2\sqrt{5}}{6-4}\\\\\\=\dfrac{\sqrt{30}-2\sqrt{5}}{2}$

Similarly , we rationalize  $\dfrac{\sqrt{5}}{\sqrt{6}-2}$ by multiplying $\sqrt{6}+2$ to the numerator and denominator  , we get

$\dfrac{\sqrt{5}}{\sqrt{6}-2}\times\dfrac{\sqrt{6}+2}{\sqrt{6}+2}\\\\\\=\dfrac{\sqrt{5}(\sqrt{6}+2)}{(\sqrt{6})^2-2^2}\\\\\\=\dfrac{\sqrt{5}(\sqrt{6})-2\sqrt{5}}{6-4}\\\\\\=\dfrac{\sqrt{30}+2\sqrt{5}}{2}$

Now , $(\dfrac{\sqrt{5}}{\sqrt{6}+2})-(\dfrac{\sqrt{5}}{\sqrt{6}-2})$

$=\dfrac{\sqrt{30}-2\sqrt{5}}{2}-(\dfrac{\sqrt{30}+2\sqrt{5}}{2})$

$=\dfrac{\sqrt{30}}{2}-\dfrac{2\sqrt{5}}{2}-\dfrac{\sqrt{30}}{2}-\dfrac{2\sqrt{5}}{2}$

$=-\sqrt{5}-\sqrt{5}=-2\sqrt{5}$

The simplified answer to the given expression is $-2\sqrt{5}$.

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Rationalise the denominator and simplify 16÷root41-5

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