If z= (1+i 1-i ), then z^4 equals. - Vidyadip

MCQ

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

A
$0$
B
$-1$
C
$2$
✓
$1$

✓

Answer

Correct option: D.

$1$

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

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