In this lesson, we are going to continue using the lowest common denominator to help us simplify expressions. The lowest common denominator or LCD has tremendous power. We're going to use it again to help us simplify complex fractions. A fraction like this one. I'll guide you through the questions you want to ask yourself to simplify a fraction like this. To get started, we want to find the lowest common denominator of all the internal fractions of our complex fraction. Complex fractions, like this one, have fractions either in the numerator or in the denominator, or in this case, both. So, let's start our path to try and simplify this crazy looking thing. What do you think is the LCD for the denominators x, 4, and 8. Write your answer here.
To find the LCD of these three denominators, we want to find the highest power of each factor that appears in any denominator. We know four is already a factor of eight, so we really just want to find the greatest common denominator for x and eight. The only common factors for x and eight are one, so we'll have x times 1 times 8, which equals 8x, our lowest common denominator.
Now that we know the LCD for this complex fraction, we can multiply the numerator and the denominator by the LCD. We want to make sure we multiply this entire numerator by 8x, and this entire denominator by 8x. We do this since we're really multiplying by a form of one. So what do you think would be the product of this multiplication? I want you to write the product for the numerator here and the product for the denominator here. Keep in mind you should have two terms for your numerator and two terms for your denominator. Good luck.
This would be the numerator and this would be the denominator, fantastic algebra skills if you figure those two out. Now, I know you might have never seen this being distributed to a fraction. Lets see how we can do that. We'll distribute the 8x to each term in the numerator and to each term in the denominator. This pare is the easiest of the multiplications we'll have 6 times 8x, which is 48x, for the second one the x's will cancel. We will have an x in the numerator and an x in the denominator. So we'll be left with 3 times 8 which equals 24. If that seems a little off to you, think of 8x as 8x divided by 1. You know x divided by x equals 1. So we're left with 3 times 8 which equals 24. For this multiplication 8 divided by 4 equals 2. So we're left with x times 2 x, which equals 2 x squared. And finally, for the last one, 1 8th of 8x would just be 1 x. 8 divided by 8 equals 1, so we're left with 1 x.
Now this is something we're familiar with again, it's just a rational expression. We don't have any fractions in the numerator and denominator. We just have a polynomial here and a polynomial here. So use your knowledge of simplifying rational expressions to figure out the simplified form. What do you think is our final answer?
Our final answer is 24 divided by x. You've really got the hang of simplifying for finding this one out. To simplify this expression, we want to factor the numerator and denominator. 24 is the greatest common factor for these two terms, and x is the greatest common factor for these two terms. When we factor, we can see that we have a 2x plus 1 in our numerator and denominator. So these cancel to make 1. So, this crazy original expression turns into 24 divided by x. Pretty nice.
Looking back at our work, we can simplify a complex fraction by following these steps. First we want to find the LCD of the internal fractions, each of these. Then we multiply our complex fraction by the LCD divided by the LCD. We can do this since this is really a form of 1. We'll change what this fraction looks like, but we won't change its value. Then we'll distribute the LCD to each term, like we did here, and then we'll simplify to get our final answer. Let's see if you can do the next one in a couple of steps.
For this complex fraction, I want you to start by just finding the LCD. What do you think it is?
The lowest common denominator for each of these fractions is k times r, or kr. Keep in mind that K is some unknown number and so is R. So the only factor that we're certain that they share is 1. So k times 1 is k and 1 x r is r. This means that the product of these factors must be the LCD.
So our next step is to multiply by the LCD divided by the LCD, or really just a form of one. So you do this multiplication. What would be the new numerator by multiplying this by kr, and what would be our new denominator, by multiplying this by kr? Good luck.
Well, we'd get 12r plus 3k for the numerator, and 12r minus 3k for the denominator. Nice algebraic skills if you got these two correct. If this gave you some trouble, that's totally okay. Let's see how we can distribute this kr to each of these terms. If we distribute kr to each of these terms, we can see that we'll have this complex fraction. Now we need to simplify the multiplication for each term. Some students are really great at doing it in their head from this step. Though, sometimes I think it's better to show this, so you can really see which factors are cancelling in a numerator and a denominator. For example, we can see that k divided by k equals 1 here, r divided by r equals 1 here, and we repeat that process in our denominator. So we're left with 12 times r, or 12r, plus 3 times k, or 3k. For the denominator, we're left with 12 times r minus 3 times k. Then, we should always ask ourselves if we can simplify our answer. I've rewritten it here so we can see it better. We don't want to make the mistake that this fraction equals 1. We have addition here, and subtraction here. Also, these are not opposite polynomials, so this fraction doesn't reduce to negative 1. The coefficient of r is positive 12 here, and positive 12 here. But, we can see that this term and this term share a common factor of 3. The same is true here. We factor a positive 3 from the numerator and from the denominator. This leaves us with 4r plus k divided by the quantity 4r minus k, our final fraction. After multiplying by the LCD over the LCD, we really want to check if we can simplify. Sometimes this isn't always our final answer.
Let's try another problem. What do you think would be the LCD for this complex fraction? Write your answer here.
The LCD would be x squared times x plus 2. Remember we want the highest power of any factor that appears in a denominator. So we know x is already a factor of x squared, so we should use x squared. And X plus two for our LCD. Another way to think about this, is to use our factor tree method to find the LCD of X squared and X plus two. We use X squared here, since we know X is already a factor of X squared. Now don't make the mistake of putting X as the common factor here. X plus two is one factor. Wgereas X squared has the factors of X and X. The only common factor for X squared and the quantity X plus two is one. This also makes sense because this X is a term inside this factor. So our LCD is X squared times one times the quantity X plus two.
Try finishing this problem out. I want you to multiply by the LCD, divided by the LCD, and then simplify your answer and put it here.
Here's our numerator and here's our denominator. Fantastic algebra skills if you got these two correct. This one was really tough, so don't worry if you didn't get it right on your first try or even at all. Let's see how we can multiply by the LCD and simplify. We know we're going to multiply by the LCD divided by the LCD. When we do that, we really just distribute this to each term in our complex fraction. This gives us this expression. Now, this is pretty hideous. So let's clean it up. We know x plus 2 divided by x plus 2 equals 1. So we're left with 3 times x squared for our first term. This next multiplication was a little tricky. We have x squared divided by x. We can only cancel 1x from this power and this denominator. This means we have to drop the squared power down to a 1. And we're left with 4 times 1x times x plus 2. I want to make sure I have it negative 4 times 1x. So I have negative 4x. Then I had times x plus 2 here. This denominator is a little bit more simple. These factors simplify to 1 and these factors also simplify to 1. So, we're left with 2 times x squared for the first term of our denominator and 1 times x plus 2 or just x plus 2 for the second term of our denominator. I need to multiply or distribute this negative 4x to the quantity x plus 2. This gives me a new numerator. For the denominator, I don't multiply. I really add, since we have addition between these two terms. Since there aren't any like terms here, I just list out each term, 2x squared plus x plus 2. And to finish out we just simply combine the like terms in our numerator. 3x squared minus 4x squared is negative x squared. So here's our final rational expression.
Okay. For our third practice problem, try simplifying this complex fraction. Write your numerator here and your denominator here.
if you got this one correct. The lowest common denominator might have been tricky, so let's find that first. We look at the denominators of 3 a squared b, and ab. We know these two denominators share a common factor of a times b. The other factor for this denominator is 3a. And the other factor for this denominator is 1. We multiply them together to get 3 times a squared times b. And remember we want the highest power of any variable or factor that appears in our denominators, 3 is the highest number factor, a squared is the highest variable factor for a And 1b or just b is the highest for, well b. Next we multiply by the LCD divided by LCD a form of one. These two factors would cancel out, leaving us with 2xy in the numerator. Here we can cancel out 1a and 1b. This means we have to drop this power down by one. Leaving us with 3a times this expression. 3 times 6 is 18 and then we just list the product of the variables. Now you might be wondering why I listed this a first instead of on the end. Usually we list the product of the variables in alphabetical order but it's certainly okay to leave this a on the end. And finally we want to simplify our rational expression. These are the factors of the numerator. And these are the factors of the denominator. We can cancel the factors that equal 1, which leaves us with a 1 in the numerator and 9 times a times x times y, in our denominator. This is why it's important to remember that 2 divided by 2 equals 1. There's still a 1 in our numerator. We want to make sure that appears in our answer. We can't simplify this anymore, since 1 is the only factor in our numerator so we're completely done.
Try simplifying this complex fraction. Write your numerator here, and your denominator here. Good luck. I'll also give you a caution. Here, we have m plus
N plus two divided by the quantity three M plus fifteen is correct. Nice work for getting either of these correct. We'll start by finding the lowest common denominator for these two fractions. We'll find the common factors for N plus five. And m plus two. Make sure that you don't put a single m here. They have a term of m, but they don't have a common factor of m. And it's because we have addition between each of these terms. Multiplying by the L-C-D divided by the L-C-D, we can see that the factors that m plus five cancel out. These equal one. Likewise, in the denominator of the complex fraction M plus 2 divided by m plus m plus 5 for our denominator. We can simplify these numbers, since they're factors that appear in the numerator and denominator. 3 9ths reduces to 1 3rd. We just divide each number by 3. Finally we distribute our 3 to get n plus 2 divided by 3m plus 15, our final answer.
Here's another complex fraction. Try simplifying it.
A squared minus 2a minus 3 divided by the quantity, a squared plus a minus 1 is correct. I hope these are getting easier. We know the LCD must be a times a minus 1. Since these are the only factors that appear in our denominators. We multiply the fraction by the LCD divided by the LCD, so we really multiply each of these terms by the number a times a minus 1. For our first term the a is reduced to 1. So we're left with a plus 3 times a minus 1. For the second term the a minus 1s cancel, leaving us with negative 4 times a. And this is why you want to be really careful to pay attention to this sign. We have negative 4 a not positive 4 a, and in our denominator we're left with a times a which is a squared, and 1 times a minus 1. We add that as our second term in the denominator. If we multiply these two together we'll get a squared plus 2a minus like terms to get negative 2a for our numerator. Now, we definitely want to try to see if we can simplify this more, but I know I can't. I can't find factors of negative 1 that sum to positive 1, so we're really done. The numerator can be factored, but again this doesn't help us out, since we can't cancel a factor that appears here with one that appears here. This is our final answer.
Here's our last complex fraction. Try simplifying it.
Three y plus two divided by y plus eight is correct. Great effort if you got this one correct. Here, the L-C-D is y plus two. These are the only factors that appear in the denominators inside of our complex fraction. We know we need to multiply by the L-C-D divided by the L-C-D. So we really just distribute this to each of our terms and our complex fraction. Next we simplify the common factors from a numerator and denominator. This leaves us with 1 times y plus 2 here, 2y here, 4 times y plus 2 in our denominator, and a negative 3y also in our denominator. We can clean up this fraction by adding our light terms here to get pro. Way to go.