The least common denominator is the lowest common multiple of the denominators.

Multiple of 27: 27, 54, 81, 108, 135, 162, 189, 216, 243, 270

Multiple of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90

The first step of finding the LCD of a set of fractions is to make sure each of the fractions are simplified.  and  are already simplified. However,  can be reduced to . This makes the problem much easier because we now only have two different denominators to work with. From here, we simply multiply each denominator by increasing integers until we get a common denominator. It is important to always increase the lower of the two denominators. For instance, we have 4 and 3 as denominators in this problem. Since 3 is lower, we will multiply it by 2, getting 6. Now we have 4 and 6. 4 is lower, so we multiply it by 2 to get 8. Now we have 8 and 6. 6 is lower, so we multiply the original denominator of 3 by 3, resulting in denominators of 8 and 9. Following this trend, we get: 12 and 9, then 12 and 12. Therefore, 12 will be the least common denominator.

While simply multiplying all of the denominators will get you a common denominator between the fractions, it does not always give you the LCD.

When finding the least common denominator, the quickest way is to multiply the numbers out.

In this case  and  are both primes and don't share any factors other than .

We can multiply them to get  as the final answer.

Another approach is to list out all the factors of each number and see which factor is in both sets first.

Notice  appears in both sets before any other number therefore, this is the least common denominator.

When finding the least common denominator, the quickest way is to multiply the numbers out.

In this case  and  share a factor other than  which is . We can divide those numbers by  to get  and  leftover.

Now, they don't share a common factor so basically multiply them out with the shared factor. Answer is .

Another approach is to list out the factors of both number and find the factor that appears in both sets first.

We can see that  appears in both sets before any other number thus, this is our answer.

When finding the least common denominator, the quickest way is to multiply the numbers out. In this case  and  share a factor other than  which is . We can divide those numbers by  to get  and  leftover. Now, they don't share a common factor so basically multiply them out with the shared factor. Answer is .

Another approach is to list out the factors of each number. The factor that appears first in both set is the least common denominator.

We see that  appears first in both sets and thus, is the least common denominator.

When finding the least common denominator, the quickest way is to multiply the expression out. In this case  and   don't share any factors other than . We can multiply this to get  as the final answer. 

Remember when foiling, you multiply the numbers/variables that first appear in each binomial, followed by multiplying the outer most numbers/variables, then multiplying the inner most numbers/variables and finally multiplying the last numbers/variables.

When finding the least common denominator, the quickest way is to multiply the expression out. In this case  and  share a factor other than  which is . If you don't see that. just break down the quadratic equation to simple factors. Remember, we need to find two terms that are factors of the c term that add up to the b term.

The quadratic becomes . By factoring out , we get  and . Just multiply the leftovers and the factored expression to get .

When finding the least common denominator, the quickest way is to multiply the numbers out. In the case of finding least common denominators among three or more numbers, it's critical there are no common factors between two of the denominators and of course all 3. This will ensure the answer will always be the least common denominator. 

Say we just multiplied the numbers out. It's basically  or . That number seems big but lets cut this in half and check   divides evenly into , , and . Lets check .  doesn't divide evenly into  so  is the answer. 

So this goes back to the statement: "In the case of finding least common denominators among three or more numbers, it's critical there are no common factors between two of the denominators and of course all 3." If I factored a , I can reduce the  and  but not the . That is ok. Now the leftover values are , , and . They only share a factor of . So let's multiply the leftover values and the factored value to get 

The first step is to find the least common denominator. In this case, it is .

Then, you convert each fraction by multiplying the first fraction by  and the second fraction by .

Once you have both fractions with a common denominator, you can add the numerators.

.

The least common denomiator (or least common multiple...same concept) is the least expression that both denominators can go into. I like to work step by step with each term. Let's start with the numerical coefficients.

The least common multiple of 12 and 60 is 60. You can figure this out by writing out multiples of 12 and 60 and seeing the first one they have in common.

Now let's move on to the a's.

There's an and an Therefore, their LCM is .

Do the same with the b's and the c's; and , respectively.

Now put those all together to get .