# Why is the following solution valid to the given complex system of equations?

The question given is - what is the value of complex number z if it satisfies the following system of equations [w represents any complex number]:

$$z^3 + \overline{w}^{7} = 0$$

$$z^5w^{11} = 1$$

Here, I understand that if one complex number = another complex number, the modulus and arguments are equal and using this logic I solved the question the following way:

$$z^3 = -\overline{w}^{7}$$

$$|z|^3 = |w|^7$$

$$7arg(w) = \pi - 3arg(z)$$

$$arg(w) = \frac{\pi - 3arg(z)}{7}$$

$$|z|^5|w|^{11} cis(5arg(z) + 11arg(w)) = 1$$

$$|z|^5|w|^{11} cis(\frac{11\pi + 2arg(z)}{7}) = 1cis(0)$$

$$|z|^5|w|^{11} = 1$$

$$|z| = |w| = 1$$

$$11\pi = -2arg(z)$$ or $$9\pi = -2arg(z)$$

$$arg(z) = -\frac\pi2$$ or $$arg(z) = \frac\pi2$$

so z = i, -i

This is a valid solution. However, the way the book did it seems faster but I don't seem to understand it. Their solution is listed below:

$$|z|^3 = |w|^7$$

$$|z|^5|w|^{11} = |1|$$

$$|z| = |w| = |1|$$

So far so good. All the steps listed are clear. But then, to find the argument they do the following -->

$$\overline{w}^{77} . w^{77} = -z^{33}.z^{-35}$$

$$z^2 = -1$$

z = +i, -i

Could someone guide me as to how they got the equations after the comment? Any help would be appreciated. Thanks!

• $$z^3 + \bar{w}^{7} = 0 \implies \bar{w}^{7} = -z^3 \implies \left(\bar w^7\right)^{11} = \left(-z^3\right)^{11} \implies \bar w^{77} = -z^{33}$$
• $$z^5w^{11} = 1 \implies w^{11} = z^{-5} \implies \left(w^{11}\right)^7 = \left(z^{-5}\right)^7 \implies w^{77} = z^{-35}$$