Question

if a and b are rational number then find the value of a and b for which(5+3√3)/(7+4√3). = a-b√3
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Answer 1

Answer :-

$\sf \dfrac{5 + 3\sqrt{3}}{7+4\sqrt{3}} = a - b\sqrt{3}$

Solving LHS part :-

$\implies\sf \frac{5 + 3\sqrt{3}}{7+4\sqrt{3}}$

Rationalizing the denominator -

$\implies\sf \Big(\dfrac{5 + 3\sqrt{3}}{7+4\sqrt{3}}\Big) \times \Big(\dfrac{7-4\sqrt{3}}{7-4\sqrt{3}}\Big)$

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$\implies\sf \dfrac{(5+3\sqrt{3})(7-4\sqrt{3})}{( 7 + 4\sqrt{3})(7 - 4 \sqrt{3})}$

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$\implies\sf \dfrac{5(7-\sqrt{3}) + 3\sqrt{3}(7-\sqrt{3})}{7^2 - (4\sqrt{3})^2}$

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$\implies\sf \dfrac{ 35 - 5\sqrt{3} + 21\sqrt{3} - 3\times 3}{49 - 48 }$

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$\implies\sf \dfrac{ 35 - 16 \sqrt{3} - 9}{1}$

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$\implies\sf 26 - 16 \sqrt{3}$

Comparing LHS and RHS :-

$\implies\sf 26 - 16 \sqrt{3} = a - b\sqrt{3}$

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• $\boxed{\sf a = 26}$
• $\boxed{\sf b = 16}$