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The coefficient of the highest power of x in the expansion of ${(x + \sqrt{x^{2} - 1})}^{8} + {(x - \sqrt{x^{2} - 1})}^{8}$ is

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256

512

128

answer is C.

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Detailed Solution

$= {(x + \sqrt{x^{2} - 1})}^{8} + {(x - \sqrt{x^{2} - 1})}^{8}$

$= 2 (8_{C_{0}} x^{8} {(\sqrt{x^{2} - 1})}^{0} + 8_{C_{2}} x^{6} {(\sqrt{x^{2} - 1})}^{2} + 8_{C_{4}} x^{4} {(\sqrt{x^{2} - 1})}^{4} + \cdot \cdot \cdot + 8_{C_{8}} x^{0} {(\sqrt{x^{2} - 1})}^{8})$

$= 256$

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