Question

If $z_1,z_2,z_3,z_4$ are 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. $1+i$

Answer 1

The correct option is A $0$

Clearly,
$z_1,z_2,z_3,z_4$ are $4^{th}$ roots of unity
$z_k=e^{i\left(\frac{2k\pi }{4}\right)}$ $[n^{th}roots=e^i\frac{k2\pi }{n}]$
Thus roots are in G.P with ratio $e^{\frac{i2\pi }{4}}$
Now, for
${z_1}^3+{z_2}^3+{z_3}^3+{z_4}^3=\frac{{z_1}^3(1-r^4)}{1-r}$
Here $r=e^{i\frac{2\pi }{4}\times 3}$
$=\frac{{z_1}^3(1-e^{i\frac{2\pi }{4}\times 3\times 4})}{1-r}$
$=\frac{{z_1}^3(1-1)}{1-r}$ $\left[\begin{array}{c}{e}^{\frac{i2\pi }{4}\times 3\times 4}\\ =e^{i6\pi }=1\end{array}\right]$
[also a property of $n^{th}$ roots of unity that $1^P+{{\alpha }_1}^P+{{\alpha }_2}^P......{{\alpha }_{n-1}}^P=0$ provided P is not a multiple of n]