Find the value of given expression

$\frac{2 (\sqrt[2]{2})}{(\sqrt[2]{5}) - (\sqrt[2]{3})}$ - $\frac{3 (\sqrt[2]{3})}{(\sqrt[2]{5}) - (\sqrt[2]{2})}$ +
$\frac{(\sqrt[2]{5})}{(\sqrt[2]{3}) - (\sqrt[2]{2})}$

$2 \sqrt[2]{10}$

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$4 \sqrt[2]{5}$

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$2 \sqrt[2]{5}$

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$4 \sqrt[2]{10}$

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Solution

The correct option is A
$2 \sqrt[2]{10}$

Rationalizing the above expression we get

$x = (\frac{2 (\sqrt[2]{2})}{(\sqrt[2]{5}) - (\sqrt[2]{3})} \times \frac{(\sqrt[2]{5} + (\sqrt[2]{3})}{(\sqrt[2]{5}) + (\sqrt[2]{3})}) - (\frac{3 (\sqrt[2]{3})}{(\sqrt[2]{5}) - (\sqrt[2]{2})} \times \frac{\sqrt[2]{5} + \sqrt[2]{2}}{\sqrt[2]{5} + \sqrt[2]{2}}) + (\frac{(\sqrt[2]{5})}{(\sqrt[2]{3}) - (\sqrt[2]{2})} \times \frac{(\sqrt[2]{3}) + (\sqrt[2]{2})}{(\sqrt[2]{3}) + (\sqrt[2]{2})})$

$x = (\frac{2 (\sqrt[2]{2})}{(\sqrt[2]{5})^{2} - (\sqrt[2]{3})^{2}} \times (\sqrt[2]{5} + (\sqrt[2]{3}) - (\frac{3 (\sqrt[2]{3})}{(\sqrt[2]{5})^{2} - (\sqrt[2]{2})^{2}} \times \sqrt[2]{5} + \sqrt[2]{2}) + (\frac{(\sqrt[2]{5})}{(\sqrt[2]{3})^{2} - (\sqrt[2]{2})^{2}} \times (\sqrt[2]{3}) + (\sqrt[2]{2}))$

$x = 2 \times (\frac{(\sqrt[2]{10} + \sqrt[2]{6})}{5 - 3}) - 3 \times (\frac{(\sqrt[2]{15} + \sqrt[2]{6})}{5 - 2}) + (\frac{(\sqrt[2]{15} + \sqrt[2]{10})}{3 - 2})$

$x = \sqrt[2]{10} + \sqrt[2]{6} - \sqrt[2]{15} - \sqrt[2]{6} + \sqrt[2]{15} + \sqrt[2]{10}$

$x = 2 \times \sqrt[2]{10}$

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