(A) The roots of the equation $z^4 = 1$ are the fourth roots of unity,which are $z_1 = 1, z_2 = i, z_3 = -1, z_4 = -i$.We need to calculate the sum $\

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(A) The roots of the equation $z^4 = 1$ are the fourth roots of unity,which are $z_1 = 1, z_2 = i, z_3 = -1, z_4 = -i$.We need to calculate the sum $\sum_{i=1}^4 z_i^3 = z_1^3 + z_2^3 + z_3^3 + z_4^3$.Substituting the values: $1^3 + (i)^3 + (-1)^3 + (-i)^3$.$= 1 - i - 1 + i = 0$.Therefore,the sum is $0$.

Class 11 Mathematics 1 Complex Numbers and Quadratic Equations: Complex Numbers De Moivre's theorem and Roots of unity

Medium

If $z_1, z_2, z_3, z_4$ are the roots of the equation $z^4 = 1$,then the value of $\sum_{i=1}^4 z_i^3$ is

A
$0$
B
$1$
C
$i$
D
$1 + i$

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1 Complex Numbers and Quadratic Equations: Complex Numbers

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De Moivre's theorem and Roots of unity

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If $z_1, z_2, z_3, z_4$ are the roots of the equation $z^4 = 1$,then the value of $\sum_{i=1}^4 z_i^3$ is

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