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Question
Evaluate the
(ix) \[\left( \sqrt{3} + \sqrt{2} \right)^6 - \left( \sqrt{3} - \sqrt{2} \right)^6\]
Solution
(ix) \[(\sqrt{3} + \sqrt{2} )^6 - (\sqrt{3} - \sqrt{2} )^6 \]
\[ = 2[^{6}{}{C}_1 (\sqrt{3} )^5 (\sqrt{2} )^1 + 6{6}{}{C}_3 (\sqrt{3} )^3 (\sqrt{2} )^3 + ^{6}{}{C}_5 (\sqrt{3} )^1 (\sqrt{2} )^5 ]\]
\[= 2[6 \times 9\sqrt{3} \times \sqrt{2} + 20 \times 3\sqrt{3} \times 2\sqrt{2} + 6 \times \sqrt{3} \times 4\sqrt{2}]\]
\[ = 2[\sqrt{6}(54 + 120 + 24)]\]
\[ = 396\sqrt{6}\]
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