If x+1x=2, find the value of x64+1x64

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Question

$If x + \frac{1}{x} = 2, find the value of x^{64} + \frac{1}{x^{64}}$

Options:

A . 32

B . 16

C . 8

D . 2

Answer: Option D
:
D

$x + \frac{1}{x} = 2$
Squaring both sides, we get $x^{2} + \frac{1}{x^{2}} + 2. x . \frac{1}{x} = 4$
So, $x^{2} + \frac{1}{x^{2}} = 4 - 2 = 2$
Again Squaring both sides, we get $x^{4} + \frac{1}{x^{4}} + 2. x^{2} . \frac{1}{x^{2}} = 4$
$\Rightarrow x^{4} + \frac{1}{x^{4}} = 4 - 2 = 2$
Similarly, $x^{8} + \frac{1}{x^{8}} = x^{32} + \frac{1}{x^{32}} = x^{64} + \frac{1}{x^{64}} = 2$

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