INSTRUCTOR: OK. Let's take a look at complex numbers in polar form if we try to take roots of them, like square roots, cubed roots, et cetera. Before we do that, I want to remind you of De Moivre's theorem here. It tells us how to raise a complex number in polar form here in green to a power n. And so you can see where the n shows up here in the formula. It's very easy, actually, to see why this is true. And we've done that before. So that's De Moivre's theorem. 

Now what we need to remember is roots, how to think of roots as exponents. So if I have the n-th root of x, that's the same thing as x to the 1/n. The root goes in the denominator here OK. And that makes sense. Root you think of being at the bottom. So the root goes in the denominator there. So we have to know those two pieces of information before we can expound on De Moivre's theorem to get roots. And so that's what we'll do now. So 

Let's say if z is r cosine theta plus i sine theta, this complex number, then what is an n-th root of z? And let's use a better word than "an." Let's use the word "one" because we'll see that it has many. 

Then one of the n-th roots of z is-- well, that would be z to the 1/n to have an n-th root of z. But z is just this complex number here in polar form. So I could take that whole thing and raise it to the 1/n power. 

And finally, applying De Moivre's theorem, I would change all my n's in De Moivre's theorem to 1 over. So now I'll have r to the 1/n times cosine of 1/n times theta plus i times the sine of 1/n times theta. And so all I've done is replaced all the n's from De Moivre's theorem above, all the purple pieces, with 1/n's because I want the roots now. 

And then if we simplify this, you get r to the 1/n. Or, if you prefer, that's the same thing as just the n-th root of this real number r times the cosine of theta over n. When I multiply 1/n by theta, I get that plus i times the sine of theta over n. So this is an n-th root of this complex number, z, r times cosine theta plus i times the sine of theta. 

So there we go. We'll see in another video with containing examples that there are more than one one square roots, there's more than one cube roots, et cetera. So we'll look at that later on.