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we're asked to divide and we're dividing 6 plus 3i by 7 minus 5i and in particular when I divide this I want to get another complex number so I want to get something you know some real number plus some imaginary number so some multiple of I so let's think about how we can do this well division is the same thing and we could rewrite this as 6 plus 3i over 7 minus 5i these are clearly equivalent dividing by something is the same thing as a rational expression where that something is in the denominator right over here and so how do we simplify this well we have a tool in our toolkit that can make sure that we don't have an imaginary or complex number in the denominator and that's the complex conjugate if we multiply both the numerator and the denominator of this expression by the complex conjugate of the denominator then we will get rid of or we will have a real number in the denominator so let's do that so let's multiply the numerator and the denominator by the conjugate of this so 7 plus 5i plus 7 plus 5i is the complex conjugate of 7 minus 5i so we're going to multiply it by 7 plus 5i over 7 plus 5i and anything divided by itself is going to be 1 assuming that you're not dealing with zero zero over zero is undefined but 7 plus 5i over 7 plus 5i is one so we're not changing the value of this but what this does is it allows us to get rid of the imaginary part in the denominator so let's multiply this out our numerator our numerator we just have to multiply every part of this complex number times every part of this complex number you can think of it as foil if you like we're really just doing the distributive property twice we have six times six is or sorry six times seven which is 42 and then we have six times 5i 6 times 5i which is 35 sorry 30 I plus 30 I and then we have 3i times 7 so that's plus 21 I we plus 21 plus 21 I and then finally we have 3i times 5i 3 times 5 is 15 but we have I times I or I squared which is negative one so it's it would be 15 times negative 1 or minus 15 so that's our numerator and then our denominator our denominator is going to be well we have we have a plus B times a minus B you could think of it that way or you could just do what we just did up here actually let's just do what we did up here so you don't have to remember that difference of squares pattern and all of that seven times seven is forty-nine let's think of it in a foil way if that is helpful for you so first we did the seven times a seven that we can do the outer terms seven times 5i is plus 35 I then we can do the inner terms negative 5i times 7 is minus 35 I these two are going to cancel out and then negative 5i times 5i is negative 25 I squared negative 25 I squared is the same thing as negative 25 times negative 1 so that is plus 25 now let's simplify them these got these guys down here cancel out our denominator simplifies to 49 plus 25 is 74 and our numerator we can add the real parts so we have a 42 and a negative 15 so let's see 42 minus 5 would be 37 minus another 10 would be 27 so that is 27 and then we add and then we're going to add our 30 I plus the 21 I so 30 plus 21 30 of something plus 21 of that's oven of that same something is going to be 51 of that something in this case that something is the imaginary unit it is I let me do that in magenta I guess that's orange so this is plus 51 I and I want to write it in the form of a plus bi the traditional complex number form so this right over here is the same thing it's the same thing as 27 over 74 27 over 74 plus plus 5 plus 51 plus 51 over 74 plus 51 over 74 plus 51 over seventy four times I and we are done I want to write that I in that same orange color and we are done we have a real part and we have an imaginary part and if this last step just confuses a little bit just remember this is the same thing if it's helpful for you we're essentially multiplying both of these terms times 1 over 74 we're dividing both of these terms by 74 we're distributing the 1 over 74 times both of these I guess is one way to think about it and that's how we got this thing over here but we have a nice real part and a nice imaginary part