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**cottontails** Let z = 3 - 3i. Express the following complex numbers in polar form.

(i) z

(ii) z^4

(iii) 1/z

With parts (ii) and (iii) of the question, would z be different from being 3 - 3i? So, for (ii), it would be (3 - 3i)^4 as being z and from that, you would have to change that result into polar form. - Is that right? However, when I was working part (i), I became a bit puzzled when I realised that theta was equal to -pi/4 or 7pi/4. So, I'm unsure as to which one to use in the overall answer.

My answer/working for part (i):

z = 3- 3i

= √(3^2) + (-3^2)

= √9 + 9

= √18

= 3√2

theta = tan^-1 (-3/3) = tan^-1 (-1) = -pi/4 or 7pi/4

z = 3√2 cis (-pi/4) = 3√2 (cos pi/4 - isin pi/4)

(But then again, couldn't the answer also be: 3√2 (cos 7pi/4 + isin 7pi/4) ?)

I'm not sure if my working is right with that either. I tried with part (ii), with getting the answer (3√2)^4 cis (-pi/4 • 4) = 324 cis (-pi) = 324 (cos pi - i sin pi) ... although, as with part (i), theta was -pi or 7pi so, I'm unsure with which theta you end up using (if that makes any sense).

As for part (iii), I'm completely clueless with how to answer that question. Although, is it right to answer the questions with z = 3√2? (As seen above, that is how I did the working out for part (ii).)

If anyone could help me with my working out and with answering all/either three parts of the questions, it would be extremely appreciated. Thanks!