Question

please answer with steps​

Answer 1

Step-by-step explanation:

(i)

$\frac{5 - 2 \sqrt{3} }{7 - 4 \sqrt{3} } \times \frac{7 + 4 \sqrt{3} }{7 + 4 \sqrt{3} } = \frac{35 + 20 \sqrt{3} - 14 \sqrt{3} - 24 }{49 - 48} \\ = > 11 - 6 \sqrt{3} = a - b \sqrt{3} \\ so \: \: \: a = 11 \: and \: b = - 6$

(ii)

$\frac{5 + \sqrt{6} }{5 - \sqrt{6} } \times \frac{5 + \sqrt{6} }{5 + \sqrt{6} } \\ = - (25 + 6 + 10 \sqrt{6} ) = - 31 - 10 \sqrt{6} = > a + b \sqrt{6} \\ so \: \: a = - 31 \: \: \: and \: \: \: b = - 10$

(i)

${ (\frac{9}{4}) }^{ \frac{3}{2} } = { {( \frac{ {3}^{2} }{ {2}^{2} } )} }^{ \frac{3}{2} } = { \frac{3}{2} }^{2 \times \frac{3}{2} } =( { \frac{3}{2}) }^{3} = \frac{27}{8}$

(ii)

$\frac{1}{ {(216)}^{ - \frac{2}{3} } } + \frac{1}{(256) {}^{ - \frac{3}{4} } } + \frac{1}{ {(243)}^{ - \frac{1}{5} } } = \frac{1}{ { {6}^{3} }^{ \times - \frac{ 2}{3} } } + \frac{ {1}^{} }{ { {4}^{4} }^{ \times - \frac{3}{4} } } + \frac{1}{ { {3}^{5} }^{ \times - \frac{1}{5} } } = \frac{1}{ {6}^{ - 2} } + \frac{1}{ {4}^{ - 3} } + \frac{1}{ {3}^{ - 1} } = {6}^{2} + 4 {}^{3} + 3 = 36 + 64 + 3 = 103$