Find the least common denominator of each set of fractions. − 1 15 y , 1 90 y -\frac{1}{15y},\frac{1}{90y}

In this problem, we are asked to find the least common denominator of the given fractions. Now, notice our fractions are -1/ 15Y and 1/ 90Y. Now, to find the least common denominator, remember, that's the same thing as just finding the least common multiple of 15Y and 90Y. So, what I need to do is find this least common denominator. And since I can clearly see I have variables in each of these denominators, I'm going to use prime factorization to find the LCD. So, let's go ahead and do that. Now, we have our denominator 15Y. And then we have our other denominator, which is 90 Y. Now, let's go ahead and break this into products of prime numbers, starting with 15 Y. 15 Y is the same thing as 3 times 5 times Y, cause we always treat these variables like prime factors. Now, what about 90 Y? Well, 90 is the same thing as 2 times 45. 45 is the same thing as 5 times 9. And then 9 is the same thing as 3 x 3, and then I can just go ahead and attach this Y. So, now we have 15 Y and 90 Y broken down into prime factors. So, what exactly do I do from here? Well, at this point, we now need to collect the unique prime factors, and this is where things can get a little bit tricky, because I'm actually going to show you something I'm gonna do. Rather than starting with the 15Y, I'm actually going to start with the 90Y. The reason why I'm starting with the 90Y is cause I can see we have a lot of factors in common here. We have a 3, a 5, and a Y. We have a 3, a 5, and a Y. So, I'm actually going to start with the 90Y because I think we'll be able to knock out more of these unique prime factors here. So, I can see we have a 2, I see we have a 5, I see we have 2 3s, and I see we have a Y. Those are the unique prime factors. Now, going up to this 15 Y, I don't need to include this 3 since we already included the 2 3s for the 90 Y. I don't need to include this 5 since we already have a 5 in 90 Y, and I don't need to include this Y since we already have a Y in 90 Y. So, all I need to do for the LCD is find the product of the unique prime factors, which is literally just going to be 2 x 5 x 3 x 3 x Y. But the thing that I noticed here is that this is going to be the exact same thing as 90Y. So, it turns out 90Y is the least common denominator, and that is the solution to the problem. And this should actually make sense, because if we take a look at these denominators, I can actually clearly see that 4 times 15Y would in fact give us 90Y. So, it makes sense that we didn't have to include any of these factors here, because 90Y is in fact the LCD. That does often happen when dealing with the least common denominator, but hopefully, while working through this prime factorization, you were able to see how this process worked and gave us this result. So I hope you found this video helpful, and let's move on.