And then -5i X 5i is -25i^2 ("negative twenty five i squared"). Let's think of it in the FOIL way, if that is helpful for you. Actually, let's just do what we did up here so you don't have to remember that difference of squares pattern and all that.) Seven times seven is forty-nine. And then our denomenator is going to be - Well, we have a plus b times a minus b. So it would be fifteen times negative one, or minus 15. But we have i times i, or i squared, which is negative one. And then finally we have three i times five i. And then we have three i times seven, so that's plus twenty-one i. And then we have six times five i, which is thirty i. We have six times seven, which is forty two. You can think of it as FOIL if you like we're really just doing the distributive property twice. Our numerator - we just have to multiply every part of this complex number times every part of this complex number. But what this does is it allows us to get rid of the imaginary part in the denominator. But seven plus five i over seven plus five i is one. And anything divided by itself is going to be one (assuming you're not dealing with zero zero over zero is undefined). So we're going to multiply it by seven plus five i over seven plus five i. Seven plus five i is the complex conjegate of seven minus five i. Let's multiply the numerator and the denominator by the conjugate of this. If we multiply both the numerator and the denominator of this expression by the complex conjugate of the denominator, then we will have a real number in the denominator. 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. 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. Well, division is the same thing - and we rewrite this as six plus three i over seven minus five i. So I want to get some real number plus some imaginary number, so some multiple of i's. And in particular, when I divide this, I want to get another complex number. And we're dividing six plus three i by seven minus 5i.
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