Question
Rationalise the denominator: $\frac{1}{\sqrt{7}+\sqrt{6}-\sqrt{13}}$
✓
Answer
$=\frac{1}{\sqrt{7}+\sqrt{6}-\sqrt{13}}$
$=\frac{1}{(\sqrt{7}+\sqrt{6})-\sqrt{13}} \times \frac{(\sqrt{7}+\sqrt{6})+\sqrt{13}}{(\sqrt{7}+\sqrt{6})+\sqrt{13}}$
$=\frac{(\sqrt{7}+\sqrt{6})+\sqrt{13}}{(\sqrt{7}+\sqrt{6})^2-\sqrt{13}^2}\left[\because a ^2= b ^2=( a + b )( a - b )\right]$
$=\frac{\sqrt{7}+\sqrt{6}+\sqrt{13}}{(7+6+2 \sqrt{42})-13}$
$=\frac{\sqrt{7}+\sqrt{6}+\sqrt{13}}{13+2 \sqrt{42}-13}$
$=\frac{\sqrt{7}+\sqrt{6}+\sqrt{13}}{2 \sqrt{42}}$
$=\frac{\sqrt{7}+\sqrt{6}+\sqrt{13}}{2 \sqrt{42}} \times \frac{\sqrt{42}}{\sqrt{42}}$
$=\frac{\sqrt{7 \times 42}+\sqrt{6 \times 42}+\sqrt{13 \times 42}}{2(\sqrt{42})^2}$
$=\frac{\sqrt{7 \times 7 \times 6}+\sqrt{6 \times 6 \times 7}+\sqrt{546}}{2 \times 42}$
$=\frac{7 \sqrt{6}+6 \sqrt{7}+\sqrt{546}}{84}$
$=\frac{1}{(\sqrt{7}+\sqrt{6})-\sqrt{13}} \times \frac{(\sqrt{7}+\sqrt{6})+\sqrt{13}}{(\sqrt{7}+\sqrt{6})+\sqrt{13}}$
$=\frac{(\sqrt{7}+\sqrt{6})+\sqrt{13}}{(\sqrt{7}+\sqrt{6})^2-\sqrt{13}^2}\left[\because a ^2= b ^2=( a + b )( a - b )\right]$
$=\frac{\sqrt{7}+\sqrt{6}+\sqrt{13}}{(7+6+2 \sqrt{42})-13}$
$=\frac{\sqrt{7}+\sqrt{6}+\sqrt{13}}{13+2 \sqrt{42}-13}$
$=\frac{\sqrt{7}+\sqrt{6}+\sqrt{13}}{2 \sqrt{42}}$
$=\frac{\sqrt{7}+\sqrt{6}+\sqrt{13}}{2 \sqrt{42}} \times \frac{\sqrt{42}}{\sqrt{42}}$
$=\frac{\sqrt{7 \times 42}+\sqrt{6 \times 42}+\sqrt{13 \times 42}}{2(\sqrt{42})^2}$
$=\frac{\sqrt{7 \times 7 \times 6}+\sqrt{6 \times 6 \times 7}+\sqrt{546}}{2 \times 42}$
$=\frac{7 \sqrt{6}+6 \sqrt{7}+\sqrt{546}}{84}$
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