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Rationalise the denominator and simplify:
\( \frac{1+\sqrt{2}}{3-2 \sqrt{2}} \)
Given:
\( \frac{1+\sqrt{2}}{3-2 \sqrt{2}} \)
To do:
We have to rationalise the denominator and simplify the given expression.
Solution:
We know that,
Rationalising factor of a fraction with denominator ${\sqrt{a}}$ is ${\sqrt{a}}$.
Rationalising factor of a fraction with denominator ${\sqrt{a}-\sqrt{b}}$ is ${\sqrt{a}+\sqrt{b}}$.
Rationalising factor of a fraction with denominator ${\sqrt{a}+\sqrt{b}}$ is ${\sqrt{a}-\sqrt{b}}$.
Therefore,
$\frac{1+\sqrt{2}}{3-2 \sqrt{2}}=\frac{(1+\sqrt{2})(3+2 \sqrt{2})}{(3-2 \sqrt{2})(3+2 \sqrt{2})}$
$=\frac{3+2 \sqrt{2}+3 \sqrt{2}+2 \sqrt{2} \times \sqrt{2}}{(3)^{2}-(2 \sqrt{2})^{2}}$
$=\frac{3+5 \sqrt{2}+4}{9-8}$
$=\frac{7+5 \sqrt{2}}{1}$
$=7+5 \sqrt{2}$
Hence, $\frac{1+\sqrt{2}}{3-2 \sqrt{2}}=7+5 \sqrt{2}$.