How can I get the absolute value of the following complex number such as $|(1-i\sqrt{2})^3|$ ?

what is right way to solve it?

  • $\begingroup$ Let $1-i\sqrt{2}=r e^{i\theta}$. get $r$ then cube it. $\endgroup$ – Anurag A May 13 '15 at 19:08
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    $\begingroup$ First calculate $|1-i\sqrt{2}|$ and then take that value to the power of $3$. $\endgroup$ – Tim B. May 13 '15 at 19:08
  • $\begingroup$ @LeBtz you can post an answer i'll accept it since I was able to solve it from what you said. $|(1-i\sqrt{2})^3| = (\sqrt{3})^3 = 3\sqrt{3} $ $\endgroup$ – Dan Revah May 13 '15 at 19:12
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    $\begingroup$ I just gave the hint. You can answer your own question now ;) $\endgroup$ – Tim B. May 13 '15 at 19:15

More generally, your question could be rephrased as: $|z^n|\stackrel{?}{=}|z|^n$. The answer is yes.

To see this, you need to switch from the Cartesian representation of complex number, where $z=a+ib$, to Polar coordinates where $z=re^{i\theta}$: $$a=r\cos\theta$$ $$b=r\sin\theta$$ $$z=a+ib=r(\cos\theta+i\sin\theta)=re^{i\theta}$$ In this representation, you have: $|z|=r$ and $\arg{z}=\theta$, i.e. $r$ is the distance from the origin to the point $z$ and $\theta$ is the angle $z$ makes with the real axis. You have then: $$|z^n|=\left|\left(re^{i\theta}\right)^n\right|=\left|r^ne^{in\theta}\right|=r^n$$ $$|z|^n=\left|re^{i\theta}\right|^n=r^n$$ $$\therefore |z^n|=|z|^n$$

And in your particular example: $$\left|(1-i\sqrt{2})^3\right|=\left|1-i\sqrt{2}\right|^3=(\sqrt{3})^3=3\sqrt{3}$$


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