Complex number calculation (to the third power)

complex numbers

I have a question that asks me to calculate $Z_1^3$ knowing that $Z_1 = 5 + 5i$

So, first I calculate by using the most straight-forward method which is $Z_1\cdot Z_1\cdot Z_1$ by using the formula $Z_1\cdot Z_2 = (a_1a_2-b_1b_2)+(a_1b_2+a_2b_1)i$

The answer is $-250+250i$ (The same as the answer on the answer key and provided by the calculator)

However, I then tried to calculate by using De Moivre's formula by converting this complex number to polar coordinates:

  • $Z_1 = p\operatorname{cis}\theta$
  • $p = \sqrt{5^2+5^2} = \sqrt{50} $
  • $\theta = \arctan(5/5) = 45^\circ$
  • $Z_1 = \sqrt{50}\operatorname{cis}45$

According to the De Movire's formula, $Z^n = p^ncis(n*\theta)$, therefore:

  • $Z_1 = \sqrt{50}^3\operatorname{cis}135^\circ$

However, there is no way this can equal to $-250+250i$, since $ \sqrt{50}^3$ is surely not a whole number.

Then I am not sure where I did wrong? I also followed the example in the book where they do the same to calculate $(1+i)^{10}$.

Best Answer

So, it turns out that I was assuming since $\sqrt{125000}$ isn't a whole number, then the result couldn't be a whole number. However, apparently, after multiplying it by $cos135$ it is!

So the answer should be

  • $cis135 = cos135+sin135$
  • $Re(Z_1) = pcos\theta = \sqrt{125000}cos135 = -250$
  • $Im(Z_1) = psin\theta = \sqrt{125000}sin153 = 250$
  • $Z_1 = -250 + 250i$

Thank you for helping me on this!

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