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Rationalisation

Simplify: 3√...

Question

Simplify:
$\frac{3 \sqrt{2} - 2 \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}} + \frac{\sqrt{8}}{\sqrt{3} - \sqrt{2}}$

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Solution

$= \frac{3 \sqrt{2} - 2 \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}} \times \frac{3 \sqrt{2} - 2 \sqrt{3}}{3 \sqrt{2} - 2 \sqrt{3}} + \frac{\sqrt{8}}{\sqrt{3} - \sqrt{2}} \times \frac{\sqrt{3} + \sqrt{2}}{\sqrt{3} + \sqrt{2}}$

$= \frac{(3 \sqrt{2} - 2 \sqrt{3})^{2}}{18 - 12} + \frac{\sqrt{8} (\sqrt{3} + \sqrt{2})}{1}$

$= \frac{30 - 12 \sqrt{6}}{6} + \frac{2 \sqrt{6} + 4}{1} = 5 - 2 \sqrt{6} + 2 \sqrt{6} + 4 = 9$

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