i am a bit stuck here.
As the title says i try to find out how to write complex numbers like for example$$ (1-i\sqrt{3})^6$$ in the normal representation$$ z = x + i*y$$
I already found out that the polar representation of complex numbers will come in handy here, but i can't make the conclusion at the moment.
How can i get from here to the polar representation? How do i get the real and imaginary part from the polar representation? If you have a hint, can you please just leave a quick post here, thanks.
Best Answer
we know $$w=\dfrac{-1+\sqrt{3}i}{2}\Longrightarrow w^2+w+1=0\Longrightarrow w^3=1$$ so $$(1-i\sqrt{3})^6=(\sqrt{3}i-1)^6=2^6\cdot\left(\dfrac{\sqrt{3}i-1}{2}\right)^6=2^6w^6=2^6\cdot 1^2=64$$