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>> I wanted to take a look at two more examples of adding sign fractions. The first thing we should notice about this example is that the denominators are not the same and then we know if that if we want to add fractions, we have to obtain a common denominator. Hopefully the least common denominator. So if you can look at the denominator of 8 and 12 and realize that the LCD would be 24, that's great. [ Pause ] We would then multiply this fraction by 3 over 3 and multiply this fraction by 2 over 2. But I also want to show how you can determine the least common denominator, using the prime factorization of the denominators. So let's rewrite these fractions, rewriting the original denominators in prime factored form. And we're also going to move these negative signs up into the numerator. So for this first fraction, we're going to write negative 3, all over the prime factorization of 8, which would be 2 times 2 times 2 plus, write the second fraction as negative 5 all over the prime factorization of 12. Well 12 is 4 times 3; 4 is 2 times 2, so this would be 2 times 2 times 3. So, again, if we're not able to determine the least common denominator is 24 by inspection, we just need to realize that in order to have a common denominator, the prime factors in the denominators must be the same. Notice how this fraction is missing a factor of 3 that the other fraction has, so we'd have to multiply this fraction by 3 over 3. And notice how the second fraction is missing one factor of 2 that this denominator has. So in order for these denominators to be the same, we have to multiply this fraction by another factor of 2, on top and on bottom. So notice how now, these fractions contain the same prime factors, and therefore, we have the least common denominator, which in both cases, is 24. Let's go ahead and rewrite this one more time. This is going to be negative 9 all over 24, plus this will be negative 10 all over 24. So the denominators will now stay the same, and we add the numerators. So negative 9 plus negative 10 is equal to negative 19, so this is our sum. But again, normally in most textbooks, you'll see the negative sign out in front of the fraction. So let's go ahead and write this as negative 19, 24. These two fractions are equivalent but this form is more common. Let's go and take a look at a second example. Again the first thing we should notice is we do have to obtain a common denominator and again, if you can look at 20 and 8 and realize that the least common denominator would be 40 -- [ Pause ] -- we could quickly multiply this first fraction by 2 over 2 and multiple the second fraction by 5 over 5. But again, I do want to show that if you can't determine the least common denominator by inspection, how you can use the prime factorization of the denominators to help. So let's go ahead and rewrite this, with the denominators in prime factored form. And we'll also move the negative here up into the numerator. So we'll have negative 3. A prime factorization of 20 would be 4 times 5, and 4 is 2 times 2, so we'll have 2 times 2 times 5 plus 1 over the prime factorization of 8, which would be 2 times 2 times 2. So now, if the prime factors of the denominators are the same, we'll have a common denominator. So now notice in order for these prime factors to be the same, this denominator will need another factor of 2, so we'll divide both the top and bottom by 2 and notice how the second fraction is missing the prime factor of 5. So multiply the second fraction by 5 over 5. Now, if we rewrite this, we'll have negative 6 all over; 2 times 2 times 5 times 2, which would be 40. Plus this would be 5, again, all over 40. So now our denominator is the same, we can go ahead and add our fractions. The denominator stays the same, and then we combine the numerators. So negative 6 plus 5 is equal to negative 1. Then again, we'll go ahead and write this, so the negative sign is out front of the fraction. And here's our sum. So this prime factorization technique to determine the LCD is not necessary if you can identify the LCD by inspection, by looking at the original denominators. But if you can't, I think it is helpful to determine the least common denominator. I hope this was helpful. ------------------------------c058722fc5a1--