complex numbers and the power of high. So let's first start with complex numbers. And basically, when we're talking about complex numbers, we're talking about numbers that are not perfect squares. Now we can understand what a perfect square is this any number that results from a number times itself. For example, square root of four is to because to comes to a sport swear of 81 is nine, because nine times nine is 81. But numbers that are not perfect square. We're gonna consider complex numbers, and we are going to write them out with a square root. So let's start with the doing a couple. So let's start with square root of 50 now. The easiest way to do this is factor out. If you want a factor out, find any factors of 50 that are perfect squares. For example, 50 is the square square, and 50 is the square of 25 times the square root of two. Now I wrote both the factors in square root cause I want to decide. Are they well, the square root of 25? It's just five. I can't find square root of two. It's going to be prompt. So that means that how I'm gonna write my complex number of square root of 50 is this going to be five, which is the perfect square, part of the factor times the square root of two. So let's do 45. Well, two factors of 45 or nine and Bob, So we want to find the square root of both of those. Well, nine is a perfect square and a square root is three. Five is not, and I can't find a square root, and it's we've got it down to a prime number, so that means that we're down to three times the square root of five. Let's look at him. Even when they are fractions, we can do this. So the square root of 11/49 Well, what that really is This the square root of 11 over the square root of 49 now square root of 49 of course, is seven square root of 11. Well, it is a complex number. It doesn't have a perfect square, but 11 is also a prime number, meaning the only two factors of 11 or 11 and one. There's no other factors, so I'm just gonna leave that one as the square root of 11. We can't factor it out anymore. Let's do the square root of 32/81. Well, this is the same as square to 32 over the square root of 81. Well, the square root of 81 is a perfect square, so it'll be nine. So let's just square root of 32. Well, two factors of 32 are going to be four and eight. Well, four is a perfect square, and it's gonna be too eight. We confined more factors of A. We want to get this down to prom. So if two factors of eight are going to be four and two again perfect for the perfect square, and it's going to be too. So what's gonna happen is I'm gonna have to times to Times Square to to and then we can simplify that to four times the square root off to We also have what we call in math imaginary numbers. And it's kind of fun to talk about imaginary numbers because we talk about imagination, things and all this. Well, imaginary numbers are very simple. There negative their negative square roots and where we come from is, for example, let's do a perfect square. Even if it's a perfect square, it can be an imaginary number. So, for example, the square root of negative four Well, we know that square root of four. Yes, to And we get that by two times two equals four. We also get it with negative two times negative. Two equals four. But there's not a number Tom's itself that will give me negative four. So that is why this is considered an imaginary number. And we write imaginary numbers out with the letter I So I was really not uses a variable in any other algebraic form except imaginary numbers. So what we're gonna do is when we find this and to find the complex number first thing we're going to do is we're gonna factor that I Well, how do you factor it out? Well, the I actually stands for the square root of negative one. So we're gonna factor out the square root of negative one, and then we'll be left with the other number. So let's start with square root of negative 80. So the first thing we're gonna do factor out square root of negative one times. And that would leave me with square root of 18. Positive 18. Well, two factors of 18 or nine and two nine is a perfect square. So that would be three. So that would leave me with three times the square root of two times the square. A negative one or three. I square root of two. Mhm. If I'm doing square root of negative 125 again, I'm most factor out square negative one which is gonna leave me with that, I and then I'm gonna leave me with square root of 125. Well out of 2225 I can factor out 25 5. 25 is a perfect square. Five five is a prime number, so I can't factor it out anymore. So that would leave me with five I times square root of five. And even if the number is a perfect square, you could still do it. So, for example, let's go back to the one we did a second ago. Square root of negative four. Well, we can still factor that out. That would be the square root of negative one times square root of four. Well, square negative one is I and square root of four is too. So my solution would be to Ah. So as we're doing these, we can actually combine and solve operations We a complex and imaginary numbers. So let's do negative two I times seven. I Well, in order to do this, we're gonna kind of treat the I like a variable and it's gonna be kind of not. So let's go ahead and multiply So negative two times seven we know is negative 14 and I times eyes. If it was a variable, would be I square. Well, we're going to simplify this anymore. If one eye is the square root of negative one, then that means that I squared is going to be square root of negative one times square root of negative one which, when we multiplied square root times itself, it just equals that number. So we would equal just negative one. So that means that negative 14 times instead of I squared, we're gonna multiply it times negative one and negative 14 times negative one is positive. 14. Let's look at this one negative 10 times negative 15 Well, before we multiply and let's go ahead and factor amount and find out what exactly we have. So we have square root of I times square root, mean square of negative, one times square root of 10 that's gonna give me I times. Now I can factor out 10. But the problem is, I can't factor out any perfect squares out of 10. So that would just I'm gonna leave it as square a 10 or I swear to 10. 15 We factor out negative one and square to 15, which would be I times again 15. Even though it has other factors, it does not have any perfect squares as factors. So I'm gonna leave it as square root of 15 or I swear to 15 so I can multiply like this. Well, that's going to give me. I squared times 10 times 15 is 150. So what we need to do is we need to separate this. Well, we've said on the previous slide that I squared is the same as square it off negative one times square it of negative one, which means that I squared equals negative one and then we need to fund square root of 150. So what we're gonna have is we're gonna have negative one times. Well, out of 150 we can factor out 25 times square to six. So that's gonna be five times square to six. So that means that negative one times five is negative five and swear to six would be my solution.