Let , be two roots of the equation x^2+(20)^1/4x+(5)^1/2=0.

Question

Let $\alpha,\beta$ be two roots of the equation $x^2+(20)^{1/4}x+(5)^{1/2}=0$. Then $\alpha^8+\beta^8$ is equal to

Accepted Answer

$\alpha+\beta=-(20)^{1/4}$, $\alpha\beta=5^{1/2}$.

$(\alpha+\beta)^2=\alpha^2+\beta^2+2\alpha\beta \implies \sqrt{20}=\alpha^2+\beta^2+2\sqrt5 \implies \alpha^2+\beta^2=0$ (since $\sqrt{20}=2\sqrt5$).

$$\alpha^8+\beta^8=(\alpha^2+\beta^2)^2-2\alpha^4\beta^4+2(\alpha\beta)^4=0+2\cdot(5^{1/2})^4=2\cdot25=50.$$

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