$x^{4}$ +1/$x^{4}$ when x-1/x =2

asked by guest
on Dec 08, 2023 at 4:27 am

You asked:

Given \(x - \frac{1}{x} = 2\) evaluate the expression \({x}^{4} + \frac{1}{{x}^{4}}\).

MathBot Answer:

\(x^{4} + \frac{1}{x^{4}} \in \left\{\frac{1}{\left(1 + \sqrt{2}\right)^{4}} + \left(1 + \sqrt{2}\right)^{4}, \left(1 - \sqrt{2}\right)^{4} + \frac{1}{\left(1 - \sqrt{2}\right)^{4}}\right\}\)


The 2 solutions to the equation are: \[\begin{aligned}x &= 1 + \sqrt{2} \approx 2.4142136\\x &= 1 - \sqrt{2} \approx -0.41421356\end{aligned}\]

By substituting this value into the expression, we find that: $$\begin{aligned}x^{4} + \frac{1}{x^{4}} &= \frac{1}{\left(1 + \sqrt{2}\right)^{4}} + \left(1 + \sqrt{2}\right)^{4}\\&= \frac{1}{\left(1 + \sqrt{2}\right)^{4}} + \left(1 + \sqrt{2}\right)^{4}\end{aligned}$$

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