If x+1/x= $\sqrt{5}$ then x^4-1/x^4=?

asked by guest
on Jan 16, 2025 at 10:54 pm



You asked:

Given \(x + \frac{1}{x} = \sqrt{5}\) evaluate the expression \({x}^{4} - \frac{1}{{x}^{4}}\).

MathBot Answer:

\(x^{4} - \frac{1}{x^{4}} \in \left\{- \frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2}\right)^{4}} + \left(- \frac{1}{2} + \frac{\sqrt{5}}{2}\right)^{4}, - \frac{1}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2}\right)^{4}} + \left(\frac{1}{2} + \frac{\sqrt{5}}{2}\right)^{4}\right\}\)

The 2 solutions to the equation are: \[\begin{aligned}x &= - \frac{1}{2} + \frac{\sqrt{5}}{2} \approx 0.61803399\\x &= \frac{1}{2} + \frac{\sqrt{5}}{2} \approx 1.618034\end{aligned}\]

By substituting this value into the expression, we find that: $$\begin{aligned}x^{4} - \frac{1}{x^{4}} &= - \frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2}\right)^{4}} + \left(- \frac{1}{2} + \frac{\sqrt{5}}{2}\right)^{4}\\&= - \frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{5}}{2}\right)^{4}} + \left(- \frac{1}{2} + \frac{\sqrt{5}}{2}\right)^{4}\end{aligned}$$

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