# how to compute $|a-ib|^2$ if given $(a-ib)^3$

If i know that, for example, $(a-ib)^3=5+4i$ - how can i compute the value of $|a-ib|^2$ ?

I can take modulus of $5+4i$ which is $\sqrt{5^2+4^2}$ but i don't know what i'm getting here.

I don't know what's a third root of a complex number here.

Any help?

Thanks.

## 1 Answer

We know that $|z\cdot w|=|z|\cdot|w|$, and so in general $|z^n|=|z|^n$.

Therefore, since you know $z^3$, you can take its modulus to get $|z^3|=|z|^3$, and raising that to the $2/3$ power gives $|z|^2$.

• so i'll get: $(\sqrt{5^2+4^2})^{2/3}$ – bob Jul 11 '15 at 11:10
• @bob: Yes, and then use power laws to fold the square root into the exponent. – Henning Makholm Jul 11 '15 at 11:11
• Do i have to? or is it fine to write that as it is? – bob Jul 11 '15 at 11:12
• @bob: As far as mathematics cares you don't "have to" do anything; you can decide to drop everything and go out in the sun to play ball instead. However, if you want to convince someone in particular that you understand power laws, then it's not advisable to give an answer in the form of a root raised to a non-integer power .... – Henning Makholm Jul 11 '15 at 11:17
• good to know, i just thought it looks simpler in that way. thank you for your nice answer! – bob Jul 11 '15 at 11:48