Let's give this problem a try. In this problem, we are told if z is equal to 1,024 times cos(\frac{\pi}{2}) + i \cdot sin(\frac{\pi}{2}), calculate the 5th root of z. And recall that the 5th root of z is the same thing as z^{1/5}. Now, we can solve this problem by recognizing that we're dealing with a complex root. So, for complex roots, what I'm going to do is write out everything that we know about these.

I know that if I want to find Z^{1/n}, that's going to equal the r value, the number out in front, to the 1/n power, and that's going to be times cis(\theta_k) where cis is a combination of the cosines and sines. Now, \theta_k is the tedious part of the problem where you have to calculate each of these angles, but I know that \theta_k = \frac{1}{n} ( \theta + 2\pi k ). We use 2\pi in case of radians. For the k values, they are going to start at 0 and go 1, 2, all the way up to n - 1.

This is \theta_k, each of the z values, and the ultimate setup. Now, I can figure out in this problem what the r values are going to be and what the k values are going to be because that's going to help me a lot. The r value is 1,024. If I want to find r^{1/n}, in this case, 1/n is 1/5, so you can see that this is going to be equal to 1024^{1/5}. The 5th root of 1,024 turns out to just be 4. This is going to be the r value we use throughout this problem.

So next, let's figure out the k values because k values go from 0 all the way up to n - 1, on which the numerator of the denominator is 5 minus 1, which is 4. So, the k values are going to be 0, 1, 2, 3, and 4, and these are the k values we're going to use in this problem.

As we have the r and we have the k, we can actually set up what this problem is going to look like. I can see that we have 5 k values, which means we're going to have 5 solutions in this problem. So we're going to have z_0, z_1, z_2, z_3, and z_4. And these are going to be all the solutions we have in this problem.

Now for z_0, it's going to follow the pattern: 4 \cdot cis(\frac{\pi}{10}), that's our first solution right there. After repeating this process for each z_k, applying the formula \theta_k = \frac{1}{5} (\frac{\pi}{2} + 2\pi k), we subsequently find each solution's angle as follows: - z_0 = 4 \cdot cis(\frac{\pi}{10}) - z_1 = 4 \cdot cis(\frac{3\pi}{10}) - z_2 = 4 \cdot cis(\frac{9\pi}{10}) - z_3 = 4 \cdot cis(\frac{13\pi}{10}) - z_4 = 4 \cdot cis(\frac{17\pi}{10})

These right here are all the solutions. This is just all the work we did to find each of these angles. I hope you found this video helpful. Thanks for watching, and please let me know if you have any questions.