Finding the complex number Find the complex number $z$ satisfying the system of equations $z^2 +
 \bar{w}^7=0$ and $z^5 w^{11}= 1$


Using Euler's form made it really complicated because we don't know the values of the modulus of z and w. Also, taking modulus on both sides of each equation didn't help too. 

Answer is $i$

Note: $w$ is not cube root of unity.
Let $|z|=a$ and $|w|=b$.

We have $$z^5 w^{11}=1 \implies |z|^5 \cdot |w|^{11}=1 \implies \color{blue}{a^5b^{11}=1} \tag 1$$

Also, 

$$z^2=-\bar{w}^{7} \implies |z|^2 = |w|^7 \implies \color{blue}{a^2=b^7} \tag 2$$

Using $(1)$ and $(2)$, we get $$|z|=|w|=1$$

Now, let $z=e^{i \theta} \quad ; \;  w=e^{i \phi}$ and proceed.
Well, you have $z^5=w^{-11}$ and $z^2=-\bar{w}^7$.

Now $-2\times 2+5=1$, so $z=(z^2)^{-2}z^5=\bar{w}^{-14}w^{-11}.$