Question

SOLVE THIS WITH STEPS
$5 + 2 \sqrt{3} \div 7 + 4 \sqrt{3} = a + b \sqrt{3}$
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Answer 1

Answer:

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Answer 2

$\tt given : - \frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } = a + b \sqrt{3} \\ \\ \tt LHS : - \\ \\ \tt = \frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } \\ \\ \sf rationalizing \\ \\ \tt = \frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } \times \frac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} } \\ \\ \tt = \frac{(5 + 2 \sqrt{3} )(7 - 4 \sqrt{3}) }{(7 + 4 \sqrt{3})(7 - 4 \sqrt{3}) } \\ \\ \tt = \frac{5(7 - 4 \sqrt{3}) + 2 \sqrt{3} (7 - 4 \sqrt{3} )}{( {7})^{2} - ( {4 \sqrt{3} })^{2} } \\ \\ \tt = \frac{35 - 20 \sqrt{3} + 14 \sqrt{3} - 24}{49 - 48} \\ \\ \tt = \boxed{11 + 34 \sqrt{3} }$

RHS :-

a + b√3

comparing LHS with RHS

➡ 11 + 34√3 = a + b√3

therefore a = 11 and b = 34

identity used while rationalizing :- (a + b) (a - b) = a² - b²