Finding common denominators

we're asked to rewrite the following two fractions as fractions with a least common denominator least common denominator so a least common denominator for two fractions is really just going to be the least common multiple of both of these denominators over here and the reason the value of doing that is then if you can if you can make these a common denominator then you can add the two fractions and we'll see that in other videos but first of all let's just find the least common multiple so the least common let me write it out because sometimes LCD could meet other things so least common common denominator denominator of these two things or is going to be the same thing as the least common multiple of the two denominators over here the least common multiple of 8 and 6 and a couple of ways to think about least common multiple you literally could just take the multiples of 8 and 6 and see what their smallest common multiple is so let's do it that way first so multiples of 6 or 6 12 18 24 30 and I could keep going if we don't find any common multiples and this out of this group here with any of the multiples in 8 and the multiples of 8 are 8 16 24 and it looks like we're done and we could keep going obviously 32 so on and so forth but I found a common multiple and this is their smallest common multiple they have other common multiples 48 and and and 72 and we can keep adding more and more multiples but their smallest common multiple their least common multiple so it's it is it is 24 another way that you could have found that least common multiple is you could have taken the prime factorization of 6 and say hey that's 2 and 3 so anything that is any the least common multiple has to have at least one 2 and 1/3 in its prime factorization in order for it to be divisible by 6 and you could have said what's the prime factorization of 8 it is 2 times 4 and 4 is 2 times 2 so in order to be divisible by 8 you have to have it least three twos in the prime factorization so to be divisible by six you have to have a two times the three and then to be divisible by 8 you have to have at least three twos we only have or you have to have two times itself three times I should say well we have one two now let's throw in a couple more so then you'll have another two and then another two so this part right over here makes it divisible by 8 and this part right over here makes it divisible by six if I take two times two times two times three that does give me 24 that gives me 24 so our least common multiple of eight and six which is also this least common denominator of these two fractions is going to be 24 so what we want to do is rewrite each of these fractions with 24 is the denominator so I'll start with 2 over 8 and I want to write that as something over 24 I want to write that as something over 24 well to get the denominator would be 24 we have to multiply it by 3 8 times 3 is 24 and so if we don't want to change the value of the fraction we have to multiply the numerator and the denominator by the same thing so let's multiply the numerator by 3 as well 2 times 3 is 6 so 2 eighths is the exact same thing as 6 24 it's to see that a little bit clearer you say look if I have two eights and if I multiply this if I multiply this times 3 over 3 that gives me 6 over 24 and this and this are the same fraction because 3 over 3 is really just 1 it's one whole so 6 to 8 is 6 24 it's and let's do the same thing with 5 6 5 6 so 5 over 6 is equal to something over 24 let me do that in a different color I'll do it in blue something over 24 to get the denominator from 6 to 24 we have to multiply it by 4 so if we don't want to change the value of 5 6 we have to multiply the numerator the denominator by the same thing so let's multiply the numerator times 4 5 times 4 is 20 5 6 is the same thing as 20 2014 we've written to eighths as 6:24 it's and we've written 5/6 as 2024 it's if we wanted to add them now we could literally just add 6 24 to 20 24 it's and I'll leave you there because they didn't ask us to actually do that