Problem with solving a complex equation

Question

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.

Answer 1

Note that

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

and for $|Z|=2$

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

Answer 2

HINT.

$$Z = X + iY$$

$$\bar Z = X - iY$$

Assuming the "bar" symbol denotes the complex conjugate.

If you multiply by $Z$ you get

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

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

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

Answer 3

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?