Let alpha and beta be the roots of x^2 - sqrt | vedclass

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(A) Given the quadratic equation $x^2 - \sqrt{2}x + 1 = 0$.Using the quadratic formula,the roots are $x = \frac{\sqrt{2} \pm \sqrt{2 - 4}}{2} = \frac{

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(A) Given the quadratic equation $x^2 - \sqrt{2}x + 1 = 0$.Using the quadratic formula,the roots are $x = \frac{\sqrt{2} \pm \sqrt{2 - 4}}{2} = \frac{\sqrt{2} \pm i\sqrt{2}}{2} = \frac{1}{\sqrt{2}} \pm i\frac{1}{\sqrt{2}}$.These can be written in polar form as $\alpha = e^{i\pi/4}$ and $\beta = e^{-i\pi/4}$.Then,$\alpha^{50} + \beta^{50} = (e^{i\pi/4})^{50} + (e^{-i\pi/4})^{50} = e^{i25\pi/2} + e^{-i25\pi/2}$.Since $e^{i25\pi/2} = e^{i(12\pi + \pi/2)} = e^{i\pi/2} = i$ and $e^{-i25\pi/2} = -i$.Therefore,$\alpha^{50} + \beta^{50} = i + (-i) = 0$.

Let $\alpha$ and $\beta$ be the roots of $x^2 - \sqrt{2}x + 1 = 0$. Then the value of $\alpha^{50} + \beta^{50}$ is:

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