If z is a complex function and ${z}^{3}=8i$ , what are all the values of z?

Pam Stokes
2022-01-18
Answered

If z is a complex function and ${z}^{3}=8i$ , what are all the values of z?

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Vivian Soares

Answered 2022-01-19
Author has **36** answers

or

So roots are

aquariump9

Answered 2022-01-20
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where

alenahelenash

Answered 2022-01-24
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Let $\omega ={e}^{\frac{2i\pi}{3}}=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$ is a root of unit. So the roots of ${Z}^{3}=1\text{}\text{are}:{\omega}^{0}=1;\omega \text{}\text{and}\text{}{\omega}^{2}=\stackrel{\u2015}{\omega}$
We have: ${z}^{3}=8i\iff {z}^{3}=(-2i{)}^{3}\iff (\frac{z}{-2i}{)}^{3}=1$
So: $\frac{z}{-2i}={\omega}^{k}k\in 0;1;2$ .
Thus The root of the equation ${z}^{3}=8i$ are:
$z=-2i{\dot{\omega}}^{k}k\in 0;1;2\text{}i.e.\text{}z=-2i\text{}\text{or}\text{}z=\sqrt{3}+i\text{}\text{or}\text{}z=-\sqrt{3}+i$

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