How can I calculate something like $(i+1)^{33}$ or similar high exponent without the use of a calculator?
$\endgroup$ 21 Answer
$\begingroup$HINT
Euler's formula $$e^{i \theta} = \cos(\theta) + i \sin(\theta)$$ and recall that any complex number can be written as $x+iy = r e^{i \theta}$.
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$\endgroup$ 5We have $1+i = \sqrt2 e^{i\pi/4}$. Hence, $$(1+i)^{33} = (\sqrt2 e^{i \pi/4})^{33} = 2^{33/2} e^{i(33 \pi/4)} = 2^{33/2} e^{i \pi/4} = 2^{16}(1+i)$$
