Question

the value of 1+✓2÷✓5+✓3+1-✓2÷✓5-✓3

Answer 1

Hey friend,
Here is the answer you were looking for:
$\frac{1 + \sqrt{2} }{ \sqrt{5} + \sqrt{3} } + \frac{1 - \sqrt{2} }{ \sqrt{5} - \sqrt{3} } \\ \\ on \: rationalizing \: we \: get \\ \\ = \frac{1 + \sqrt{2} }{ \sqrt{5} + \sqrt{3} } \times \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} - \sqrt{3} } + \frac{1 - \sqrt{2} }{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} + \sqrt{3} } \\ \\ using \: the \: identity \\ (a + b)(a - b) = {a}^{2} - {b}^{2} \\ \\ = \frac{1 \times \sqrt{5} - 1 \times \sqrt{3} + \sqrt{2} \times \sqrt{5} - \sqrt{2} \times \sqrt{3} }{ {( \sqrt{5}) }^{2} - {( \sqrt{3} )}^{2} } + \frac{1 \times \sqrt{5} + 1 \times \sqrt{3} - \sqrt{2} \times \sqrt{5} - \sqrt{2} \times \sqrt{3} }{ {( \sqrt{5}) }^{2} - {( \sqrt{3}) }^{2} } \\ \\ = \frac{ \sqrt{5} - \sqrt{3} + \sqrt{10} - \sqrt{6} }{5 - 3} + \frac{ \sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6} }{5 - 3} \\ \\ = \frac{ \sqrt{5} - \sqrt{3} + \sqrt{10} - \sqrt{6} + \sqrt{5} + \sqrt{3} - \sqrt{10} - \sqrt{6} }{2} \\ \\ = \frac{2 \sqrt{5} - 2 \sqrt{6} }{2} \\ \\ = \sqrt{5} - \sqrt{6}$

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