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Question
Evaluate the
(vi) \[\left( 2 + \sqrt{3} \right)^7 + \left( 2 - \sqrt{3} \right)^7\]
Solution
(vi) \[(2 + \sqrt{3} )^7 + (2 - \sqrt{3} )^7 \]
\[ = 2[ ^{7}{}{C}_0 \times 2^7 \times (\sqrt{3} )^0 + ^{7}{}{C}_2 \times 2^5 \times (\sqrt{3} )^2 + ^{7}{}{C}_4 \times 2^3 \times (\sqrt{3} )^4 +^{7}{}{C}_6 \times 2^1 \times (\sqrt{3} )^6 ]\]
\[ = 2[128 + 21 \times 32 \times 3 + 35 \times 8 \times 9 + 7 \times 2 \times 27]\]
\[ = 2[128 + 2016 + 2520 + 378]\]
\[ = 2 \times 5042 = 10084\]
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