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Suppose that \(z\) is a complex number such that  \(z^4 = \frac{64}{5}-\frac{48}{5}i\). Find \(|z|\).

Any help would be greatly appreciated!

 Feb 2, 2020
 #2
avatar+109723 
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I put the answer into Wolfram|Alpha and it says the answer is 2.

https://www.wolframalpha.com/input/?i=z%5E4%3D64%2F5-48%2F5i+find+%7Cz%7C

 

I have forgotten how to do it myself though.

 Feb 2, 2020
 #3
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The magnitude of the cn on the RHS is \(\displaystyle \sqrt{\left(\frac{64}{5}\right)^{2}+\left(\frac{48}{5}\right)^{2}}=16, \)

so the magnitude of z is the fourth root of 16, which is 2.

 Feb 3, 2020
 #4
avatar+202 
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Wait, so are you saying that the magnitude of \(z\) is the fourth root of the magnitude of \(z^4\)? Neat, will have to investigate to see why that is true. Thanks Guest!

Impasta  Feb 4, 2020

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