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Question
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
Options
1
−1
0
none of these
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Solution
1
\[\text {Let } z = \frac{1 + i}{1 - i}\]
Rationalising the denominator:
\[z=\frac{1 + i}{1 - i}\times\frac{1 + i}{1 + i}\]
\[\Rightarrow z = \frac{1 + i^2 + 2i}{1 - i^2}\]
\[\Rightarrow z = \frac{2i}{2}\]
\[ \Rightarrow z = i\]
\[\Rightarrow z^4 = i^4 \]
\[\text { Since} i^2 = - 1,\text { we have }: \]
\[ \Rightarrow z^4 = i^2 \times i^2 \]
\[ \Rightarrow z^4 = 1\]
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