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  4. Evaluate: (1+i)6+(1-i)3.

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Q. Evaluate : $\left(1+i\right)^{6}+\left(1-i\right)^{3}$.

Complex Numbers and Quadratic Equations

Solution:

$\left(1+i\right)^{6}=\left\{\left(1+i\right)^{2}\right\}^{3}$
$=\left(2i\right)^{3}=8i^{3}=-8i$
and $\left(1-i\right)^{3}=1-i^{3}-3i+3i^{2}$
$=1+i-3i-3$
$=-2-2i$
$\therefore \, \left(1+i\right)^{6}+\left(1-i\right)^{3}$
$=-8i-2-2i$
$=-2-10i$