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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!

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1 Answer 1

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The equation is derived by multiplying the following:

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

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

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