Solve the following


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

(a) 1

(b) −1

(c) 0

(d) none of these


(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:

$\Rightarrow z^{4}=i^{2} \times i^{2}$

$\Rightarrow z^{4}=1$



Leave a comment


Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now