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Q. The smallest positive integer $n$ , for which $\left(1+i\right)^{2n}=\left(1-i\right)^{2n}$ is

Complex Numbers and Quadratic Equations

Solution:

We have, $\left(1+i\right)^{2n} =\left(1-i\right)^{2n}$
$\Rightarrow \,\left(\frac{1+i}{1-i}\right)^{2n} =1$
$\Rightarrow \,\left(i\right)^{2n}=1$ which is possible if $n = 2$
$\left(\because\,i^{4}=1\right)$