Simplify the following fractions to lowest terms. 224 x 3 y 4 z 2 24 x y 4 z 2 \frac{224x^3y^4z^2}{24xy^4z^2}

Here we have a problem that asks us to simplify the following fraction to lowest terms. We have 224 x cubed times y 4 times z 2/24 x times y 4 times Z2. So we've got a lot that we're dealing with here. And so because personally, I don't feel too confident in just dealing with these numbers right off the bat, I'm going to go ahead and break this all down into prime factors. So what I'm going to start by doing is breaking down the numerator into prime factors. Now, 224. What I recognize is that's going to be the same thing as 2 times 112. So, I'm literally just dividing that by 2 to go ahead and get that answer. Now, what's going to give us 112 here? Well, I know that 60 plus 60 would be 120, so it's going to be something between 50 and 60 here. Well, it turns out the answer we're looking for is 256. So 2 times 56 would give us 112. Now, what is 56? Well, 56, I actually know happens to be 7 times 8.8 is the same thing as 2 times 4, and 4 is the same thing as 2 times 2. So, this is what happens when we break down all of 224 into prime numbers, but we're not even quite done yet because we still have these variables we have to deal with. We have X times X times X. We have Y times Y times Y times Y because we have. Three X's, we have 4 Y's, X to the 3rd and Y to the 4th. And then we have this Z squared or Z to the 2nd, which is going to be Z times Z. Now the numerator is completely broken down and wow, that's a lot of numbers and variables. But it's going to be really satisfying when we're able to cancel and reduce this later. So now let's go ahead and break down the denominator. Well, 24 is the same thing as 2 times 1212 is the same thing as 2 times 6, and 6 is the same thing as 2 times 3. So that's as broken down as we're going to get it. Then we have times X, and then we have times Y times Y times Y times Y because we have Y 4th, and then we have Z times Z. So, not as bad. But notice we now have everything fully broken down into prime factors, and I am going to mention here, if you were feeling a little bit more confident in what you were grouping, you are more than welcome to just go ahead and partially factor these and then cancel greater factors if you want to. But personally for me, I don't have that level of confidence when dealing with these really big numbers. So that's why I decided to break everything down here. But hopefully, as you're doing these problems, you're getting more and more comfortable with this factoring process. So what I can tell is that 2 of the 2's up here will cancel with 2 of the 2's in the denominator. I can see that we have another 2 that's going to cancel in the denominator. I don't see any 3s in the numerator, so nothing here is going to cancel. So it looks like we've canceled all the numbers that we can. Now, let's move on to the letters or variables. Well, I can see one of these X's is going to cancel here, and I can also see 4 of the Y's will cancel across the board, the numerator and the denominator. Then I can see 2 of the Z's will cancel in the numerator and the denominator. So what exactly are we going to have left over after all of this? Well, what I can see is that we have 7 x 2 x 2. Now, 2 times 2 is 4, and 4 times 7 is 28. So we end up with 28 for our numbers, and I see that we have two X's being multiplied, which gives us, which gives us X squared, and that's what we end up with in the numerator. Now, in the denominator, I can see that we have a 3, but do we have anything else? Well, it looks like all of our variables canceled, so all we're going to have is this 3. So 28 x 2/3 is going to be the final simplified answer to this problem, and that's how you can solve it. Hope you found this helpful.