Question

If $z=\left(\frac{1+i}{1-i}\right)$, then $z^4$ equals

• A. 1
• B. -1
• C. 0
• D. none of these

Answer 1

The correct option is A

1

Let $z=\left(\frac{1+i}{1-i}\right)$

Rationalising the denominator,

$z=\frac{1+i}{1-i}\times \frac{1+i}{1+i}\Rightarrow z=\frac{1+i^2+2i}{1-i^2}\Rightarrow z=\frac{1-1+2i}{1+1}\Rightarrow z=\frac{2i}{2}\Rightarrow z=i\Rightarrow z^4=i^4\text{Since}{i}^2=-1,\text{We have}:\Rightarrow z^4=i^2\times i^2\Rightarrow z^4=1$