Question

Hey, Am I talking to the maths Pro? If yes! Answer the question there ...The no 1. and 2. ..Thanks pro

Answer 1

$\textbf{\huge{\underline{ Hello Mate! }}}$

$1. \: \frac{ {x}^{a(b - c)} }{ {x}^{b(a - c)} } \div {( \frac{ {x}^{b} }{ {x}^{a} }) }^{c} \\ = \frac{ {x}^{ab - ac} }{ {x}^{ab - bc} } \times \frac{ {x}^{ac} }{ {x}^{bc} } \\ = {x}^{ab - ac - ab + bc} \times {x}^{ac - bc} \\ = {x}^{ - ac + bc + ac - bc} = {x}^{0} = 1$

$\textsf{\red{ Hence Proved }}$

$2. \: \frac{4}{ \sqrt{5} + \sqrt{3} } + \frac{43}{3 \sqrt{7} + 2 \sqrt{5} } - \frac{51}{3 \sqrt{7} + 2 \sqrt{3} } \\ \\ = \frac{4}{ \sqrt{5} + \sqrt{3} } \times \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} - \sqrt{3} } \\ = \frac{4( \sqrt{5} - \sqrt{3} )}{5 - 3} = 2 \sqrt{5} - 2 \sqrt{3} \\ \\ = \frac{43 }{3 \sqrt{7} + 2 \sqrt{5} } \times \frac{3 \sqrt{7} - 2\sqrt{5} }{3 \sqrt{7} - 2 \sqrt{5} } \\ = \frac{43(3 \sqrt{7} - 2 \sqrt{5} )}{63 - 20} = 3 \sqrt{7} - 2 \sqrt{5} \\ \\ = \frac{51}{3 \sqrt{7} + 2 \sqrt{3} } \times \frac{3 \sqrt{7} - 2 \sqrt{3} }{3 \sqrt{7} - 2 \sqrt{3} } \\ = \frac{51(3 \sqrt{7} - 2 \sqrt{3} )}{63 - 12} = 3 \sqrt{7} - 2 \sqrt{3} \\ \\ Arranging \: all \: we \: get \\ \\ = 2 \sqrt{5} - 2 \sqrt{3} + 3 \sqrt{7} - 2 \sqrt{5} - (3 \sqrt{7} - 2 \sqrt{3} ) \\ = 2 \sqrt{5} - 2 \sqrt{3} + 3 \sqrt{7} - 2 \sqrt{5} - 3 \sqrt{7} + 2 \sqrt{3} \\ = 0$

Hence answer is 0.

Have great future ahead!