Sum of roots of the equation $(z - 1)^{4} = 16$ is

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Solution

We have,
$(z - 1)^{4} = 16$
$\Rightarrow z - 1 = (16)^{1 / 4} \cdot (1)^{1 / 4}$
$\Rightarrow z - 1 = 2 \cdot e^{i 2 n π / 4}, n = 0, 1, 2, 3$
$∴ z - 1 = 2 i, - 2, - 2 i, 2$
$\Rightarrow z = 1 + 2 i, - 1, 1 - 2 i, 3$
Therefore, sum of roots $= 4$

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