X^4+X^2+1=0 then X^4+1/X^4=?

asked by guest
on Nov 14, 2024 at 10:36 am



You asked:

Given \({X}^{4} + {X}^{2} + 1 = 0\) evaluate the expression \({X}^{4} + \frac{1}{{X}^{4}}\).

MathBot Answer:

\(X^{4} + \frac{1}{X^{4}} \in \left\{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)^{4} + \frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)^{4}}, \frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)^{4}} + \left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right)^{4}, \frac{1}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2}\right)^{4}} + \left(\frac{1}{2} - \frac{\sqrt{3} i}{2}\right)^{4}, \left(\frac{1}{2} + \frac{\sqrt{3} i}{2}\right)^{4} + \frac{1}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2}\right)^{4}}\right\}\)

\(i\) is the imaginary unit, defined as \(i^2 = -1\).

By substituting this value into the expression, we find that: $$\begin{aligned}X^{4} + \frac{1}{X^{4}} &= \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)^{4} + \frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)^{4}}\\&= \left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)^{4} + \frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right)^{4}}\end{aligned}$$

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