$x^{4}$ +1/$x^{4}$ when x-1/x =2
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: $$\begin{aligned}x^{4} + \frac{1}{x^{4}} &= \frac{1}{\left(1 + \sqrt{2}\right)^{4}} + \left(1 + \sqrt{2}\right)^{4}\\&= \frac{1}{\left(1 + \sqrt{2}\right)^{4}} + \left(1 + \sqrt{2}\right)^{4}\end{aligned}$$
asked 7 months ago
active 7 months ago