This problem is really interesting because we are asked to perform multiple operations. We are given 2 complex numbers both in polar form, \( z_1 \) and \( z_2 \). We are asked to find \( z_1 \) divided by \( z_2 \), all raised to the 3rd power. So we're going to need to combine multiple skills that we've learned to solve this and to get our final solution. Let's just go ahead and go over this example to see what happens. Now the way that I'm going to do this is to break things up into 2 steps. My first step is going to be to deal with everything inside these parentheses, which is finding this quotient or basically dividing. Now my second step is going to be, once I've divided these 2, to take whatever result that I get and to raise it to the 3rd power. So I'll deal with the division first and the power second. So let's go ahead and start with step 1.
For step 1, I need to find what \( z_1 \) over \( z_2 \) is. Now to do this, we have discussed what these rules are for division. What you want to do is divide the \( r \) values and then subtract the angles. Now going up here, I can see that our \( r \) values are 4 and one half. So for \( z_1 \) divided by \( z_2 \), we're going to have 4 divided by 1 half, and then that's going to be cis and then the angles subtracted. Now, I can see that this angle here is 25 degrees. So gonna have 25 degrees minus the angle for \( z_2 \) which I can see is 10 degrees. So this right here is what we end up getting when we plug everything in. Now at this point it just turns into reducing everything and simplifying. Now, 4 divided by 1 half, we can flip this fraction and bring it to the top. That's the same thing as 4 times 2. And 4 times 2 is going to give us 8. So we're going to have 8 cis, and by the way this is also like dividing 4 by 0.5 that should give you 8 as well. So we're gonna have 8 cis, and then we have 25 minus 10, which is 15 degrees. So this is what happens when we divide \( z_1 \) and \( z_2 \), and that is what we end up getting. Now this is not the final solution because recall that this is just the first step. We've dealt with the first step here. Now what we need to do next is we need to raise this to the power of 3. So I'm gonna do that over here.
So we'll have \( z_1 \) divided by \( z_2 \), and then that's all gonna be raised to the 3rd power. And to do this, well, we figured out what \( z_1 \) divided by \( z_2 \) is. So what I can do is apply De Moivre's theorem to figure out what this is raised to the third power. So we're going to have 8, and recall we need to take the \( r \) value and raise it to the 3rd. So we'll have \( 8^3 \), and that's going to be cis. Then we need to take whatever value we have here and multiply it by the angle of 15 degrees that we calculated. So we're gonna have this number, 3 multiplied by 15 degrees. So this is what we have. Now from here, once again, our solution is going to be found by simplifying. Now first off, I need to deal with this \( 8^3 \). \( 8^3 \) is 8 times 8 times 8, which you can put on a calculator. This should come out to 512. So we're gonna have 512, and that's going to be cis 3 times 15. 3 times 15 is 45 degrees, and this is what we get. So this right here would be the final solution to the problem. That's going to be \( z_1 \) divided by \( z_2 \), all raised to the 3rd power. So, that is how you can solve these types of situations where you have multiple operations on a complex number in polar form. You first want to deal with whatever is inside the parentheses, and then deal with any kind of exponents or anything else outside of the parentheses. Hope you found this video helpful. Thanks for watching.