One of the roots of the equation (x+1)^4 + 81 | vedclass

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(C) Given the equation $(x+1)^4 + 81 = 0$. Let $y = x+1$,then $y^4 = -81$. We can write $-81 = 81 e^{i(\pi + 2k\pi)}$ for $k = 0, 1, 2, 3$. Taking the

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$-1 + 3\left(\frac{1+i}{\sqrt{2}}\right)$

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(C) Given the equation $(x+1)^4 + 81 = 0$. Let $y = x+1$,then $y^4 = -81$. We can write $-81 = 81 e^{i(\pi + 2k\pi)}$ for $k = 0, 1, 2, 3$. Taking the fourth root,$y = 3 e^{i(\frac{\pi + 2k\pi}{4})}$. For $k=0$,$y = 3 e^{i\pi/4} = 3(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}) = 3(\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}}) = \frac{3(1+i)}{\sqrt{2}}$. Since $x = y - 1$,we have $x = -1 + \frac{3(1+i)}{\sqrt{2}}$. Comparing this with the options,option $C$ is $-1 + 3\left(\frac{1+i}{\sqrt{2}}\right)$.

One of the roots of the equation $(x+1)^4 + 81 = 0$ is

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