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Let $w^4=16 j$. Which of the following cannot be a value of $w$ ?

  1. $2 e^{\frac{j 2 \pi}{8}}$
  2. $2 e^{\frac{j \pi}{8}}$
  3. $2 e^{\frac{j5 \pi}{8}}$
  4. $2 e^{\frac{j9 \pi}{8}}$
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The given equation is $w^4 = 16j$. Rewriting the right side into polar form we get $16j = 16 e^{j\frac{\pi}{2}}$.

By comparing, we can see that $w = 2e^{j\frac{\pi}{8}+k\frac{\pi}{2}}$ where k is an integer.

So the possible values of w are: $2 e^{j\frac{\pi}{8}}$ $2 e^{j\frac{5\pi}{8}}$ $2 e^{j\frac{9\pi}{8}}$ $2 e^{j\frac{13\pi}{8}}$

From the given options, $2 e^{j\frac{2\pi}{8}}$ cannot be a value of $w$._

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