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Let \(z = \frac{(-11 + 13i)^3 \cdot (24 - 7i)^4}{3 + 4i},\) and let \(w = \frac{\overline{z}}{z}.\) Compute |w|. 

 
 Aug 5, 2022
 #1
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After some extremely long computations, |w| works out to 2.

 
 Aug 5, 2022
 #2
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|w| = 1

 
 Aug 5, 2022
 #3
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We don't even need to know the value of z! Absolutely unnecessary information!

 

|w|^2 = w(cong(w))

Multiplying the conjugate of w in the second equation by the value of w given in the second equation. This gives us |w|^2 = 1

|w| = 1 as magnitudes are always nonnegative. 

 
 Aug 5, 2022

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