complex numbers Now, in order to divide complex numbers, What you're gonna do is these you're gonna be set up as fractions, and the first thing you're gonna do is you're going to multiply. Both numerator and the denominator by the complex conjugate off the denominator. So what I mean is our denominator is three plus 51 three plus five. The complex conjugate is basically What we're gonna do is we're gonna change the sign of our complex number in there, which is the five I. So we're gonna actually have three minus five I and that is actually what we're going to be multiplying it by. So when we divide con complex numbers, that's how we're going to divide. So we'll multiply this Tom's three minus five I on both the top and the bottom. So when I do that, I'm going to kind of basically use distributive property. So that would be to negative two items. Three would be negative. Six I and negative two I times negative five. I would be 10. Ah, squared and over will simplify that. Just a minute. Use and full for three plus five I and three minus five. I we would have 93 times three is nine. Then we would do three times negative five. I would be negative. 15 I plus 15 I and our last turns would be negative. 25 I squared, which we know is the same as negative. 25 times negative one because I squared is negative one. So this would be 25 so I can combine. Negative. 15 and 15 would actually give me, um those would actually cancel out. So what I'm left with would be non plus 25 would be 34 on the top. We can simplify that some more. So we have negative six I plus 10. I squared again that would actually be naked. Six I Plus we know I squared is negative one. So that was simplified to negative six I minus 10 over 34 and then we're going to break this down even further. So we're gonna have that's gonna give me negative 6/34 I and we'll add that to negative 10 over 34 and I can actually simplify both of those fractions. Negative. 6/34 would be negative. 3/17. I minus 5/70. So my solution would be that. So let's do a couple of others. Let's do three. I plus oh, divided by two plus four I So we're gonna multiply by the conjugate. So that's gonna be three three I over two plus four I, Tom's Tu minus four I over to minus four I while using order of operations that would be a six. I'm sorry. Using distributive property would be six I minus 12. I squared over using full. We'd have to when these air full for a denominator. So that would actually let me do it going this way. So that would be two times two is four negative. Eight I eight I and then you'd have 16. I squared. So what you're gonna be left with would be four plus 16. I alright, negative 16. I square. So what we have when we kind of simplify this out? Let's simplify both of those. So we have six I which that's not gonna change minus 12, and we know that I squared is the same as negative one over four, minus 16. Again I squared is the same as negative one. So simplify that that's gonna give me six I plus 12 over four plus 16, which is six I. We'll do this below it. Six I plus 12/20. Now we need to break this down even more, and we're gonna do our numbers that don't have the variable first. So let's break it down. We'll have 12/20 because that's plus six I over 20. So let's simplify those fractions. So that 12/20 that's going to simplify to 3/5. Plus, we're gonna let her. I kind of be to the side of this fraction and 6/20 will simplify to three 10. So my solution is three fists plus three tents. Let's do one more. Let's do five plus I over to I Well, for this one are Kaja get is gonna be negative too. I So we're just kind of switching the sign and our denominator of a complex number, so we'll multiply. So that's gonna be negative. T and I minus two. I squared over negative four square. So let's simplify that one even more so we have negative t and I minus two and we know I squared is negative. One over negative four. Again, I squared. It's the same as negative one. So we have negative 10 I plus two over positive four. So let's break this down. So we'll have to over four plus negative 10 I or negative 10/4 I so we can simplify both. Those fractions to over four reduces toe one half. Since our negative, our 10 is negative, let's just put minus and 10/4 reduces to five over to I. So our solution is one half minus five half.