if x^3+1/x^3=2 then x^6+1/x^6=?
You asked:
Given \({x}^{3} + \frac{1}{{x}^{3}} = 2\) evaluate the expression \({x}^{6} + \frac{1}{{x}^{6}}\).
MathBot Answer:
\(x^{6} + \frac{1}{x^{6}} \in \left\{2, \frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)^{6}} + \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)^{6}, \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)^{6} + \frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)^{6}}\right\}\)
\(i\) is the imaginary unit, defined as \(i^2 = -1\).
By substituting this value into the expression, we find that: $$\begin{aligned}x^{6} + \frac{1}{x^{6}} &= 1^{6} + \frac{1}{1^{6}}\\&= 2\end{aligned}$$