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Multiplication of Surds

If x = 9 - ...

Question

If $x = 9 - 4 \sqrt{5}$ then, find the value of $x^{4} + \frac{1}{x^{4}}$

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Solution

We have,
$\begin{matrix} x = 9 - \sqrt{5} x + \frac{1}{x} = 9 - 4 \sqrt{5} + \frac{1}{9 - 4 \sqrt{5}} = 9 - 4 \sqrt{5} + 9 + 4 \sqrt{5} = 18 N o w, {(x + \frac{1}{x})}^{2} = 324 x^{2} + \frac{1}{x^{2}} = 322 x^{4} + \frac{1}{x^{4}} + 2 = 103684 x^{4} + \frac{1}{x^{4}} = 103682 \end{matrix}$

Hence, this is the answer.

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