Question

If $z=\left(\frac{1+i}{1-i}\right)$, then z⁴ equals
(a) 1
(b) −1
(c) 0
(d) none of these

Answer 1

(a) 1
$\mathrm{Let}z=\frac{1+i}{1-i}$
^{Rationalising the denominator:
$z=\frac{1+i}{1-i}\times \frac{1+i}{1+i}$
$\Rightarrow z=\frac{1+i^2+2i}{1-i^2}$
$\Rightarrow z=\frac{1-1+2i}{1+1}$
$$\begin{gathered} \Rightarrow z=\frac{2i}{2} \\ \Rightarrow z=i \end{gathered}$$
$$\begin{gathered} \Rightarrow z^4=i^4 \\ \mathrm{Since}{i}^2=-1,\mathrm{we}\mathrm{have}: \\ \Rightarrow z^4=i^2\times i^2 \\ \Rightarrow z^4=1 \end{gathered}$$}