Rationales the denominator and simplify: 5+2 3 7+4 3 - Vidyadip

Question

Rationales the denominator and simplify: $\frac{5+2\sqrt3}{7+4\sqrt3}$

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Answer

$\frac{5+2\sqrt3}{7+4\sqrt3}$
Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor
$7-4\sqrt3$ $=\frac{\big(5+2\sqrt3\big)\big(\sqrt3-\sqrt2\big)}{\big(\sqrt3+\sqrt2\big)\big(\sqrt3-\sqrt2\big)}$
As we know, $(\text{a}+\text{b})(\text{a}-\text{b})=(\text{a}^2-\text{b}^2)$
$=\frac{\big(5+2\sqrt3\big)^2\big(7-4\sqrt3\big)}{49-48}$
$=35-20\sqrt3+14\sqrt3-24=11-6\sqrt3$

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