Question:
If $z=\left(\frac{1+i}{1-i}\right)$, then $z^{4}$ equals
(a) 1
(b) −1
(c) 0
(d) none of these
Solution:
(a) 1
Let $z=\frac{1+i}{1-i}$
Rationalising the denominator:
$z=\frac{1+i}{1-i} \times \frac{1+i}{1+i}$
$\Rightarrow z=\frac{1+i^{2}+2 i}{1-i^{2}}$
$\Rightarrow z=\frac{1-1+2 i}{1+1}$
$\Rightarrow z=\frac{2 i}{2}$
$\Rightarrow z=i$
$\Rightarrow z^{4}=i^{4}$
Since $i^{2}=-1$, we have:
$\Rightarrow z^{4}=i^{2} \times i^{2}$
$\Rightarrow z^{4}=1$