Currently studying for my calculus exam when I stumbled upon this example:
Find the real and imaginary parts of the following
$$z=(1+i)^{100}$$
Following the answer to this problem it's first stated that $1+i$ can be written in polar form as $\sqrt{2}e^{i\frac{\pi }{4}}$ which I totally get. Proceeding by rewriting it as:
$$z=\sqrt{2}^{100} * (e^{i\frac{\pi}{4}})^{100}$$
Then we get to the parts where I'm totally lost at the moment.
$$2^{50}*e^{i2\pi} = 2^{50}*e^{i\pi} = -2^{50}$$
Where the real part would be equal to $-2^{50}$ and the imaginary part to be equal to $0$ which I can see in the answer given.
However, the last line of simplification is what confuses me. Could someone explain what is done?
Thanks!
Best Answer
What about if we start with $$(1+i)^2=1+2i-1=2i.$$ Then $$(1+i)^{100}=[(1+i)^2]^{50}=(2i)^{50}=2^{50}(i^2)^{25}=2^{50}(-1)^{25}=-2^{50}.$$ So the real part is $-2^{50}$ and the imaginary part is $0$.