Question

Find (1-i)^6. Express the result in rectangular form.

Answer 1

Do you know how to expand the power?

Answer 2

You can also change to polar form, then do the sixth power, and came back to the rectangular form.

Answer 3

I don't know neither ways, can you explain the most simple way?

Answer 4

The most simple way is the polar form,
\[z=r_\theta \] As the power is,
\[z^n=r^n_{n\theta}\]

Answer 5

And it is related to rectangular form,
\[z=r(\cos\theta+i\sin\theta)\]

Answer 6

First find r
\[r=|z|=\sqrt{1^2+1^2}=\sqrt{2}\] Then find the argument
\[\theta=\arctan(Im(z)/Re(z))=\arctan(-1)=225º\]

Answer 7

Now, we have,
\[z=\sqrt{2}_{225}\]So the power is,
\[z^6=(\sqrt{2})^6_{6\cdot225}=8_{1350}\]Now find the rectangular form,
\[z=8(\cos(1350)+i\sin(1350 ))=-8i\]

Answer 8

That should be the answer if my lines are correct.

Answer 9

Oh ok thanks again

Answer 10

Sorry, it should be
\[z=8i\]
because the correct angle was 315 not 225.

Answer 11

ok thanks!