Rationales the denominator and simplify: 1+ 2 3-2 2 - Vidyadip

Question

Rationales the denominator and simplify: $\frac{1+\sqrt2}{3-2\sqrt2}$

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Answer

$\frac{1+\sqrt2}{3-2\sqrt2}$
Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor
$3+2\sqrt2$ $=\frac{\big(1+\sqrt2\big)\big(3+2\sqrt2\big)}{\big(3-2\sqrt2\big)\big(3+2\sqrt2\big)}$
As we know, $(\text{a}-\text{b})(\text{a}-\text{b})=(\text{a}^2-\text{b}^2)$
$=\frac{\big(1+\sqrt2\big)\big(3+2\sqrt2\big)}{9-8}$
$=3+2\sqrt2+3\sqrt2+4=7+5\sqrt2$

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