Question

please answer this ........

Answer 1

$\frac{7 \sqrt{3} }{ \sqrt{10} + \sqrt{3} } - \frac{2 \sqrt{5} }{ \sqrt{6} + \sqrt{5} } - \frac{3 \sqrt{2} }{ \sqrt{15} + 3 \sqrt{2} } \\ \\rationalize \: the \: denominator \\ \\ \frac{7 \sqrt{3} ( \sqrt{10 } - \sqrt{3} )}{( \sqrt{10} + \sqrt{3} )( \sqrt{10} - \sqrt{3} )} - \frac{2 \sqrt{5}( \sqrt{6} - \sqrt{5} )}{( \sqrt{6} + \sqrt{5} )( \sqrt{6} - \sqrt{5}) } - \frac{3 \sqrt{2}( \sqrt{15} - 3 \sqrt{2}) }{( \sqrt{15} + 3 \sqrt{2} )( \sqrt{15} - 3 \sqrt{2} )}$

$using \: the \: identity \: - - \\ (a - b)(a + b) = {a}^{2} - b {}^{2} \\ simplify \: the \: product \\ \\ \frac{7 \sqrt{3} ( \sqrt{10} - \sqrt{3} )}{7} - \frac{2 \sqrt{5} ( \sqrt{6} - \sqrt{5} )}{1} - \frac{3 \sqrt{2}( \sqrt{15} - 3 \sqrt{2} )}{ - 3}$

$reduce \: the \: fractions \\ \\ we \: get \\ \\ \sqrt{3} (10 - \sqrt{3} ) - 2 \sqrt{5} ( \sqrt{6} - \sqrt{5} ) - ( - \sqrt{2} ( \sqrt{15} - 3 \sqrt{2} )) \\ \\ mutiply \: through \: brackets \\ \\ \sqrt{30} - 3 - 2 \sqrt{30} + 10 - ( - \sqrt{30} + 6) \\ \\ \sqrt{30} - 3 - 2 \sqrt{30} + 10 + \sqrt{30} - 6$

$collect \: like \: terms \\ \\ \sqrt{30} - 2 \sqrt{30} + \sqrt{30} - 3 - 6 + 10 \\ \\ - \sqrt{30} + \sqrt{30} - 3 - 6 +10\\ \\ 0 +1 \\ \\ 1$