Find the value of a and b if $\frac{5 + 2 \sqrt{3}}{7 + 4 \sqrt{3}} = a + b \sqrt{3}$ .

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Solution

$L . H . S = \frac{5 + 2 \sqrt{3}}{7 + 4 \sqrt{3}}$
$= \frac{(5 + 2 \sqrt{3}) (7 - 4 \sqrt{3})}{(7 - 4 \sqrt{3}) (7 + 4 \sqrt{3})}$
$= \frac{35 - 20 \sqrt{3} + 14 \sqrt{3} - 24}{7^{2} - (4 \sqrt{3})^{2}}$
$= \frac{11 - 6 \sqrt{3}}{49 - 48} = 11 - 6 \sqrt{3}$
$∴ 11 - 6 \sqrt{3} = a + b \sqrt{3}$
On Comparing both sides
$a = 11$
$b = - 6$

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