Evaluate: frac{sqrt{2}}{sqrt{2 + sqrt{3}} - s | vedclass

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Question

(B) Let the expression be $X = \frac{\sqrt{2}}{\sqrt{2 + \sqrt{3}} - \sqrt{2 - \sqrt{3}}}$.Rationalizing the denominator by multiplying the numerator

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$1$

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(B) Let the expression be $X = \frac{\sqrt{2}}{\sqrt{2 + \sqrt{3}} - \sqrt{2 - \sqrt{3}}}$.Rationalizing the denominator by multiplying the numerator and denominator by $(\sqrt{2 + \sqrt{3}} + \sqrt{2 - \sqrt{3}})$:$X = \frac{\sqrt{2}(\sqrt{2 + \sqrt{3}} + \sqrt{2 - \sqrt{3}})}{(2 + \sqrt{3}) - (2 - \sqrt{3})}$$X = \frac{\sqrt{4 + 2\sqrt{3}} + \sqrt{4 - 2\sqrt{3}}}{2\sqrt{3}}$Since $4 + 2\sqrt{3} = (\sqrt{3} + 1)^2$ and $4 - 2\sqrt{3} = (\sqrt{3} - 1)^2$:$X = \frac{(\sqrt{3} + 1) + (\sqrt{3} - 1)}{2\sqrt{3}}$$X = \frac{2\sqrt{3}}{2\sqrt{3}} = 1$.

Evaluate: $\frac{\sqrt{2}}{\sqrt{2 + \sqrt{3}} - \sqrt{2 - \sqrt{3}}}$

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