Question

Simplify
please help make anyone ..

Answer 1

Identity used :

$(a + b)(a - b) = {a}^{2} - {b}^{2}$

Rationalizing the Denominator of

$\frac{2 \sqrt{6} }{ \sqrt{2} + \sqrt{3} }$

we get,

$\frac{2 \sqrt{6} }{ \sqrt{2} + \sqrt{3} } \times \frac{ \sqrt{2} - \sqrt{3} }{ \sqrt{2} - \sqrt{3} } \\ \\ = \frac{2 \sqrt{ {2}^{2} \times 3} - 2 \sqrt{ {3}^{2} \times 2} }{ {( \sqrt{2}) }^{2} - {( \sqrt{3} )}^{2} } \\ \\ = \frac{2 \times 2 \sqrt{3} - 2 \times 3 \sqrt{2} }{2 - 3} \\ \\ = \frac{4 \sqrt{3} - 6 \sqrt{2} }{ - 1} \\ \\ = - 4 \sqrt{3} + 6 \sqrt{2} \\ \\ \bf = 6 \sqrt{2} - 4 \sqrt{3}$

Rationalizing the denominator of

$\frac{6 \sqrt{2} }{ \sqrt{6} + \sqrt{3} }$

we get,

$\frac{6 \sqrt{2} }{ \sqrt{6} + \sqrt{3} } \times \frac{ \sqrt{6} - \sqrt{3} }{ \sqrt{6} - \sqrt{3} } \\ \\ = \frac{6 \sqrt{ {2}^{2} \times 3 } - 6 \sqrt{6} }{ {( \sqrt{6} )}^{2} - {( \sqrt{3} )}^{2} } \\ \\ = \frac{6 \times 2 \sqrt{3} - 6 \sqrt{6} }{6 - 3} \\ \\ = \frac{12 \sqrt{3} - 6 \sqrt{6} }{3} \\ \\ = \frac{3(4 \sqrt{3} - 2 \sqrt{6} )}{3} \\ \\ \bf = 4 \sqrt{3} - 2 \sqrt{6}$

Rationalizing the denominator of

$\frac{8 \sqrt{3} }{ \sqrt{6} + \sqrt{2} }$

we get,

$\frac{8 \sqrt{3} }{ \sqrt{6} + \sqrt{2} } \times \frac{ \sqrt{6} - \sqrt{2} }{ \sqrt{6} - \sqrt{2} } \\ \\ = \frac{8 \sqrt{ {3}^{2} \times 2} - 8 \sqrt{6} }{ {( \sqrt{6}) }^{2} - {( \sqrt{2} )}^{2} } \\ \\ = \frac{8 \times 3 \sqrt{2} - 8 \sqrt{6} }{6 - 2} \\ \\ = \frac{24 \sqrt{2} - 8 \sqrt{6} }{4} \\ \\ = \frac{4(6 \sqrt{2} - 2 \sqrt{6}) }{4} \\ \\ \bf = 6 \sqrt{2} - 2 \sqrt{6}$

Putting the values we get,

$(6 \sqrt{2} - 4 \sqrt{3} ) + (4 \sqrt{3} - 2 \sqrt{6} ) \\ \: \: \: \: \: \: - (6 \sqrt{2} - 2 \sqrt{6} ) \\ \\ = 6 \sqrt{2} - 4 \sqrt{3} + 4 \sqrt{3} - 2 \sqrt{6} \\ \: \: \: \: \: \: \: \: \: \: \: - 6 \sqrt{2} + 2 \sqrt{6} \\ \\ \bf = 0$