How do I find $(1+i)^{100}$ without expanding $(1+i)$ 100 times?
Is there a quicker way to do this?
The hint was to find the modulus and argument of $1+i$ which I've got as $\sqrt{2}$ and $\pi/4$ but I'm not sure what to do from here.
$\endgroup$5 Answers
$\begingroup$$$(1+i)^2=2i\qquad \Longrightarrow\qquad (1+i)^4=-4.$$
$\endgroup$ 6 $\begingroup$Hint: $$(1+i)=\sqrt{2}e^{\frac{i\pi}{4}}$$
$\endgroup$ $\begingroup$Hint:
$1+i=\sqrt{2} e^{i\pi/4}$
Therefore, $(1+i)^{100}= [\sqrt{2} e^{i\pi/4}]^{100}=\sqrt{2}^{100}e^{i100\pi/4}=[{{2^{0.5}}}]^{100}e^{i(24\pi+\pi)}=2^{50}e^{i\pi}=2^{50}(-1)=\boxed{-2^{50}}$
Your job: how did we get $1+i=\sqrt{2} e^{i\pi/4}$?
To do this kind of problem in general, look at the polar form of a complex number:
$\endgroup$ $\begingroup$You can use De Moivre's formula, $1+i=\sqrt{2}(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4})$, so $(1+i)^{100}=(\sqrt{2})^{100}(\cos 25\pi+i \sin 25\pi)$
$\endgroup$ $\begingroup$The ( at least a ) key is that the powers are periodic (argument-wise) .One way of doing Complex multiplication is by doing it geometrically, you multiply two numbers by squaring their respective moduli and by adding their respective arguments. Notice that $1+i$ lies on the line $y=x$, so that its argument is $\pi/4$. What happens if you go around $\frac {2k\pi}{\pi/4}$ times? The key is that $\pi/4$ is "commensurate" to $2\pi$, meaning it divides exactly (an integer number of times) into $2 \pi$.
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