Question

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

• A. $2e^{j\frac{2\pi}{8}}$
• B. $2e^{j\frac{\pi}{8}}$
• C. $2e^{j\frac{5\pi}{8}}$
• D. $2e^{j\frac{9\pi}{8}}$

Hint:

For $n$-th roots of a complex number $re^{j\theta}$, the solutions are $\sqrt[n]{r}\; e^{j(\theta+2k\pi)/n}$ for $k=0,1,.....,n-1$. Always check which angles appear in the set.

Answer 1

The Correct Option is A

Step 1: Express RHS in polar form.
$16j = 16e^{j\pi/2}$.
Step 2: Find fourth roots.
If $w^4 = 16e^{j\pi/2}$, then: \[ w = \sqrt[4]{16}\; e^{j\left(\frac{\pi/2 + 2k\pi}{4}\right)}, k=0,1,2,3 \] \[ \Rightarrow w = 2e^{j\left(\frac{\pi}{8} + \frac{k\pi}{2}\right)}, k=0,1,2,3 \] Step 3: Compute values.
- For $k=0$: $w=2e^{j\pi/8}$.
- For $k=1$: $w=2e^{j5\pi/8}$.
- For $k=2$: $w=2e^{j9\pi/8}$.
- For $k=3$: $w=2e^{j13\pi/8}$.
Step 4: Compare with options.
Options (B), (C), (D) match the valid roots.
Option (A) $2e^{j2\pi/8}=2e^{j\pi/4}$ is not in the list.
\[ \boxed{2e^{j\frac{2\pi}{8}} \;\; \text{cannot be a value of $w$}} \]