In complex number we have real and imaginary part mixed and \(\sqrt{-1}\) is the basic unit and denoted by \(i\). In the given question we have to find value of k for which the equation will be valid.
How many integers \(k\) are there for which \((1-i)^k=2^k\) ?
(A) One
(B) None
(C) Two
(D) More than one.
ISI entrance B. Stat. (Hons.) 2003 problem 5
Complex numbers
6 out of 10
Challenges and thrills of pre-college mathematics
The complex number \((1-i)\) can be rationalized by multiplying numerator and denominator by \(1+i\).
And we will get
\((1-i)=(1-i)\frac{(1+i)}{1+i}=\frac{2}{1+i}\)
Now we will have \((\frac{2}{1+i})^k=2^k\)
so, \((1+i)^k=1\), this is only possible when k=0,
So \(k\) can have only one value, The option (A) is correct.
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