Column I
Column II (one of the value of z)

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Solution

A) $z^{4} - 1 = 0 \Rightarrow z^{4} = 1 = (cos 0 + i sin 0)$
$\Rightarrow z = {(cos 0 + i sin 0)}^{\frac{1}{4}} = (cos 0 + i sin 0)$
B) $z^{4} + 1 = 0 \Rightarrow z^{4} = - 1 = (cos π + i sin π)$
$\Rightarrow z = {(cos π + i sin π)}^{\frac{1}{4}} = (cos \frac{π}{4} + i sin \frac{π}{4})$
C) $i z^{4} + 1 = 0 \Rightarrow z^{4} = i = (cos \frac{π}{2} + i sin \frac{π}{2})$
$\Rightarrow z = {(cos \frac{π}{2} + i sin \frac{π}{2})}^{\frac{1}{4}} = (cos \frac{π}{8} + i sin \frac{π}{8})$
D) $i z^{4} - 1 = 0 \Rightarrow z^{4} = - i = (cos (- \frac{π}{2}) + i sin (- \frac{π}{2}))$
$\Rightarrow z = {(cos (- \frac{π}{2}) + i sin (- \frac{π}{2}))}^{\frac{1}{4}} = (cos (- \frac{π}{2}) + i sin (- \frac{π}{2})) = (cos \frac{π}{2} - i sin \frac{π}{2})$

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\to A a n d \to B

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