Question

please solve it for me​

Answer 1

Question :-

$\sf{\frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } = a + b \sqrt{3} } \\ \sf{find \: a \: and \: b \: }$

Answer :-

→ a = 11 and b = -6

Solution :-

Taking Left hand side ,

$\rightarrow \: \frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{ 3} } \\ \\ \sf{ multiply \: and \: divided \: by \: 7 - 4 \sqrt{3} } \\ \: \\ \rightarrow \: \frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } \times \frac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} } \\ \\ \because \boxed{ \underline{ \: \sf{{x}^{2} - {y}^{2} = (x - y)(x + y)} }}\\ \\ \rightarrow \: \frac{(5 + 2 \sqrt{3})(7 - 4 \sqrt{3}) }{ {7}^{2} - {(4 \sqrt{ 3} )}^{2} } \\ \\ \rightarrow \: \frac{35 - 20 \sqrt{3} + 14 \sqrt{3} - 8 \times 3 }{49 - 48} \\ \\ \rightarrow \: \frac{35 - 24 - 6 \sqrt{3} }{1} \\ \\ \rightarrow \sf{11 - 6 \sqrt{3} \: \: \: \: \: }$

now comparing with Right hand side, then we get,

→ a = 11 and B = -6

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