apply complex and imaginary numbers. So let's start with some adding and subtracting. So let's do six minus four I plus one plus three. I now is clear that we need to kind of keep in mind that what? We're really still kind of combining like terms with this, even if they are imaginary numbers. So what's gonna happen, though, is we're going to treat this like two imaginary numbers we've got. Our numbers that are not basically are constants. So we're gonna have six plus one and we'll add that to are are very coefficients in front of the I would be negative four plus three times s So basically, I've kind of factor the eye out of both of those and we're gonna add just the coefficients. And that's why I said it's very similar to combining like terms. Well, six plus one is seven negative. Four and three is negative one and then we bring her I down. So our solution is seven minus one. I Well, subtraction is gonna be done very much the same way. Let's go ahead and start with our numbers that are not with the eyes. So we have three minus five monness or add And then we have negative two minus negative. Four. I we did this minus and negative because we have the minus in between or two sons. So that's gonna be three. Minus five is negative. Two plus negative. Two plus four is too I So our solution is negative. Two plus two I so adding and subtracting. You basically kind of keep the same rules as you would as if you were combining light tarps. So let's move into multiplying. Well, multiplying is very similar to another method we've used before. It's gonna be like multiplying a binomial times a binomial. So we're going to use the foil method to do so. So that means I'm going to multiply my first terms, which would be three times four my outside terms, which would be three times six I My inside terms would be negative. Five I times four and my last terms would be negative. Five I times six I. So when I simplify those that would be 12 and then I would have 18. I negative 20 I and then on have negative 30 I square now. Right here We have an issue. Well, not an issue, but it's another actually, instead of a regular quadratic or by nominal or polynomial that we've ever done here we have a value for I remember that I is the square root of negative one. So I squared is squaring negative one times square in a negative one. Which means that I squared is actually just negative one. So that means that we have negative 30 times negative one which equals positive. 30. So that means when I write my solution, I have 12 plus 18 I minus 20 I plus 30. So let's combine like terms. So we have 12 and 30 is gonna give me 42 and 18 I minus 20. I is negative two I So my solution is 40 to minus two I So let's multiply these also use them full. So for our first terms would be one times negative. One are outside would be one times four I are inside is gonna be two i times negative one and our last would be too I times four I So these would be negative one for I negative to I and then a I squared Well, we know I squared is negative one so that would be eight times negative one, which would be negative. Eight. So let's come put those together. So we have negative one plus four. I minus two, I minus eight. And when I combine like terms, I'd have negative nine plus two. I

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apply complex and imaginary numbers. So let's start with some adding and subtracting. So let's do six minus four I plus one plus three. I now is clear that we need to kind of keep in mind that what? We're really still kind of combining like terms with this, even if they are imaginary numbers. So what's gonna happen, though, is we're going to treat this like two imaginary numbers we've got. Our numbers that are not basically are constants. So we're gonna have six plus one and we'll add that to are are very coefficients in front of the I would be negative four plus three times s So basically, I've kind of factor the eye out of both of those and we're gonna add just the coefficients. And that's why I said it's very similar to combining like terms. Well, six plus one is seven negative. Four and three is negative one and then we bring her I down. So our solution is seven minus one. I Well, subtraction is gonna be done very much the same way. Let's go ahead and start with our numbers that are not with the eyes. So we have three minus five monness or add And then we have negative two minus negative. Four. I we did this minus and negative because we have the minus in between or two sons. So that's gonna be three. Minus five is negative. Two plus negative. Two plus four is too I So our solution is negative. Two plus two I so adding and subtracting. You basically kind of keep the same rules as you would as if you were combining light tarps. So let's move into multiplying. Well, multiplying is very similar to another method we've used before. It's gonna be like multiplying a binomial times a binomial. So we're going to use the foil method to do so. So that means I'm going to multiply my first terms, which would be three times four my outside terms, which would be three times six I My inside terms would be negative. Five I times four and my last terms would be negative. Five I times six I. So when I simplify those that would be 12 and then I would have 18. I negative 20 I and then on have negative 30 I square now. Right here We have an issue. Well, not an issue, but it's another actually, instead of a regular quadratic or by nominal or polynomial that we've ever done here we have a value for I remember that I is the square root of negative one. So I squared is squaring negative one times square in a negative one. Which means that I squared is actually just negative one. So that means that we have negative 30 times negative one which equals positive. 30. So that means when I write my solution, I have 12 plus 18 I minus 20 I plus 30. So let's combine like terms. So we have 12 and 30 is gonna give me 42 and 18 I minus 20. I is negative two I So my solution is 40 to minus two I So let's multiply these also use them full. So for our first terms would be one times negative. One are outside would be one times four I are inside is gonna be two i times negative one and our last would be too I times four I So these would be negative one for I negative to I and then a I squared Well, we know I squared is negative one so that would be eight times negative one, which would be negative. Eight. So let's come put those together. So we have negative one plus four. I minus two, I minus eight. And when I combine like terms, I'd have negative nine plus two. I

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