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I'm having trouble solving this equation $$Z^3=-4i\bar Z$$ I need to find Z, I've tried multiplying the equation by Z but still couldnt solve it.

I'll be glad for help.

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Note that

$$Z^3=-4i\bar Z\implies |Z^3|=4|Z|\implies |Z|=0 \quad \lor \quad |Z|=2$$

and for $|Z|=2$

$$Z^3=-4i\bar Z\iff Z^4=-4iZ\bar Z=-16i \implies Z=2(-i)^\frac14$$

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  • $\begingroup$ Thank you for your comment! $\endgroup$ – Elyasaf755 Feb 26 '18 at 16:52
  • $\begingroup$ Can i ask how did you calculate |Z^3|=4|Z|? $\endgroup$ – Elyasaf755 Feb 26 '18 at 16:53
  • $\begingroup$ I get it. Very nice thank you so much!! $\endgroup$ – Elyasaf755 Feb 26 '18 at 16:55
  • $\begingroup$ @user534957 I've taken the modulus both sides. $\endgroup$ – user Feb 26 '18 at 16:55
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I'd try taking the modulus of both sides: $|Z|^3=4|Z|$, from which $|Z| = 0$ or $2$.

For the case $|Z|=2$, we have $Z=2e^{i\theta}$.

Can you progress from there?

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HINT.

$$Z = X + iY$$

$$\bar Z = X - iY$$

Assuming the "bar" symbol denotes the complex conjugate.

If you multiply by $Z$ you get

$$(X + iY)^4 = 4i(X^2 + Y^2)$$

$$(X + iY)^4 = 4i (X + iY)(X - iY)$$

$$(X + iY)^3 = 4i(X - iY)$$

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  • $\begingroup$ How do I find z with this? $\endgroup$ – Elyasaf755 Feb 26 '18 at 16:46
  • $\begingroup$ And thank you for your cimment i really appreciate it $\endgroup$ – Elyasaf755 Feb 26 '18 at 16:47

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