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Question
Evaluate the
(x) \[\left\{ a^2 + \sqrt{a^2 - 1} \right\}^4 + \left\{ a^2 - \sqrt{a^2 - 1} \right\}^4\]
Solution
(x) \[\left\{ a^2 + \sqrt{a^2 - 1} \right\}^4 + \left\{ a^2 - \sqrt{a^2 - 1} \right\}^4 \]
\[ = 2[ ^{4}{}{C}_0 ( a^2 )^4 (\sqrt{a^2 - 1} )^0 +^{4}{}{C}_2 ( a^2 )^2 (\sqrt{a^2 - 1} )^2 + ^{4}{}{C}_4 ( a^2 )^0 (\sqrt{a^2 - 1} )^4 ]\]
\[ = 2[ a^8 + 6 a^4 ( a^2 - 1) + ( a^2 - 1 )^2 ]\]
\[ = 2[ a^8 + 6 a^6 - 6 a^4 + a^4 + 1 - 2 a^2 ]\]
\[ = 2 a^8 + 12 a^6 - 10 a^4 - 4 a^2 + 2\]
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