The expression in the image shows the fourth root of a complex number $16i$.

To solve for the fourth root of $16i$, we can follow these steps:

Step 1: Convert $16i$ to polar form

The complex number $16i$ has no real part, and its imaginary part is 16. We can express it in polar form as $r(\cos \theta + i \sin \theta)$, where:

• $r = |16i| = 16$ (the modulus),
• $\theta$ is the argument of the complex number. Since $16i$ lies along the positive imaginary axis, we have $\theta = \frac{\pi}{2}$.

Thus, in polar form: $16i = 16(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2})$

Step 2: Apply De Moivre’s Theorem

The fourth root of a complex number is given by: $\text{Root} = r^{1/4} \left( \cos \frac{\theta + 2k\pi}{4} + i \sin \frac{\theta + 2k\pi}{4} \right)$ where $k = 0, 1, 2, 3$ to find the four roots.

For $16i$, we have $r = 16$ and $\theta = \frac{\pi}{2}$.

Now, we compute the fourth roots for each $k$:

• $r^{1/4} = 16^{1/4} = 2$
• The arguments are: $\theta_k = \frac{\frac{\pi}{2} + 2k\pi}{4} = \frac{\pi + 8k\pi}{8}$

Now calculate the four roots for $k = 0, 1, 2, 3$:

1. For $k = 0$: $\theta_0 = \frac{\pi}{8}, \quad 2\left(\cos \frac{\pi}{8} + i \sin \frac{\pi}{8}\right)$
2. For $k = 1$: $\theta_1 = \frac{5\pi}{8}, \quad 2\left(\cos \frac{5\pi}{8} + i \sin \frac{5\pi}{8}\right)$
3. For $k = 2$: $\theta_2 = \frac{9\pi}{8}, \quad 2\left(\cos \frac{9\pi}{8} + i \sin \frac{9\pi}{8}\right)$
4. For $k = 3$: $\theta_3 = \frac{13\pi}{8}, \quad 2\left(\cos \frac{13\pi}{8} + i \sin \frac{13\pi}{8}\right)$

Conclusion:

These four expressions give the four distinct fourth roots of $16i$. Would you like to see the numeric evaluations or need further clarification on any part?

Related Questions:

1. How do you convert a complex number to polar form?
2. What are the properties of the argument of a complex number?
3. Can you calculate the square roots of a complex number using a similar method?
4. How would you find the cube roots of a complex number?
5. What is the general form of De Moivre's Theorem?

Tip: When dealing with roots of complex numbers, always convert them to polar form to simplify calculations.