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if u get this, u can understand the whole concept.........

\[i ^{5}=i ^{4+1}= i ^{4}*i ^{1}= 1* i=i\]

Again, I don't want to list the cases. Please, help me or tell me "No choice for you" :)

tried the exponential form? so \(\large e^{i \ n\frac{\pi}{4}}\) is handy notation

\[i^n\equiv i^{n\pmod{4}}\]

also, notice the answers will be equally spaced on the unit circle, starting at i

Show me, please.

all the red lines are of unit length ^

oh, you convert to cis, right?

Irish posted the form you should use

I got it. Thanks a lot, friends :)

Yes, Sir.

with the number \(1+i\), the length won't stay the same when you rise it to a power

\[z= (1+i)^6\]
\(|z| = \sqrt{2} \implies |z^6| = |z|^6 = (\sqrt{2})^6\)

How to find the simpliest form of (1+i)^6 ?

I meant the expanding form.

Like if we have \(z = (1+i)^2 = 2i\) ,then \(|z|= 2\), but how to work if it is z ^6 or z^8?

one sec
for magnitude, we just use the property \(|z^n| = |z|^n\)

do you want to expand (1+i)^6 just for finding the magnitude
or for some other reason ?

That is why I don't want to use cis. :)

oh ok, are you allowed to use binomial thm ?