Simplifying $z^3 e^{i\pi/3} +1 = 0 $

Question

Given $$z^3 e^{i\pi/3} +1 = 0 $$ We have, $ z^3 = e^{i2\pi/3} $

I get

$$ e^{i\pi/3}z^3 = -1 $$

$$ z^3 = \frac{-1}{e^{i\pi/3}} $$ $$ z^3 = -e^{i2\pi/3} $$ instead

May I know how did we arrived at $z^3$ ?

Answer 1

Using Euler Formula

$$e^{i\theta}=\cos\theta+i\sin\theta$$

put $\theta=\pi$ to get

$$e^{i\pi}=-1$$

$$e^{i\pi/3}z^3 +1 = 0$$

$$e^{i\pi/3}z^3 =-1$$ $$e^{i\pi/3}z^3 =e^{i\pi}$$

$$z^3 =\frac{e^{i\pi}}{e^{i\pi/3}}=e^{i(\pi-\pi/3)}=e^{i2\pi/3}$$ And you can also verify using same formula that whatever you got is not wrong it's just different way to represent it!

Answer 2

Hint

One of the most beautiful formula in mathematics is $$\Large e^{i \pi}+1=0$$ So for your equation, $$e^{i\pi/3}z^3 +1 = e^{i\pi/3}z^3 -e^{i \pi}=0$$

I am sure that you can take from here.