Intro to adding rational expressions with unlike denominators

- What I want to do in this video is really make sure that we feel comfortable manipulating algebraic expressions that involve fractions. So we'll start with some fairly straightforward ones. So let's say that I had, let's say I had A over B plus C over D, and if I actually wanted to add these things, so it is just one fraction, how would I do that? Well, what we could do is, we could find a common denominator. Well, over here, we don't know what B is, we don't know what D is, but we know a common denominator is just going to be B times D. That is going to be a common multiple of B and D. So we could rewrite this as two fractions, with the common denominator BD, so, plus, BD, actually, let me color code it a little bit. So A over B is going to be the same thing as what over BD? Well, to get BD, I multiplied the denominator by D, so let me multiply the numerator by D as well, then I haven't changed the value of the fraction, I'm just multiplying by D over D. So this is going to be A times D over B times D. Notice I could divide the numerator and the denominator by D, and I'm going to get back to A over B. And then we can look at the second fraction, C over D, to go from D to BD, we multiplied by B and so, if I multiply the denominator by B, if I don't want to change the value of the fraction, I have to multiply the numerator by B as well. So let's multiply the numerator by B as well, and it's going to be BC, BC. BC over BD. This is C over D. So what I have here in magenta, this fraction is equivalent to this fraction. I just multiplied it by D over D, which we can assume is one, if we assume that D is not equal to zero, and then, if we just multiply C over D times one, which is the same thing as B over B, if we assume that B is not equal to zero, then this fraction and this fraction are equivalent. Now, why did I go through all of this trouble? Well, now, I have a common denominator, so I can add these two fractions. So what's this going to be? Well, common denominator is BD, so let me just, so the common denominator is BD, and I could just add the numerators, just like you would've done if these were numbers, if this wasn't an algebraic expression. So this is going to be, this is going to be AD plus BC, all of that over BD.