In order to find the least common denominator, you must find the least common multiple of all three numbers.  For this problem, the least common multiple of all three numbers is 16 (divisible by 2, 4, 8 , 1, and 16). 

The lowest common denominator looks at the denominator (bottom number) of all of the fractions and finds the smallest number that all of the numbers divide into.

Of those numbers the largest is 18.  First check to see if all the numbers divide into 18.  3 and 6 do, but 4 and 12 do not.  Multiply 18 by 2 and get 36.  Check to see if all the numbers divide into 36.  3 and 6 still do.  4 and 12 do now. therefore. 36 is the lowest common denominator.

All of the denominators must be able to divide into the same number. First, see if the three smaller numbers (3, 4, 12) divide into the largest number (15)—NO. Then check the multiples of the largest number to see if the lower numbers divide into it:

15 * 2 = 30 (NO)

15 * 3 = 45 (NO)

15 * 4 = 60 (YES!)

Finding the common denomenator of  yields a result of

The least common multiple of 3, 10, and 12 is 60. 60 is divisible by all three numbers (60/3 = 20, 60/10 = 6, and 60/12 = 5). Therefore, you could convert these fractions to 20/60, 25/60, and 42/60.

Find the largest number that divides into both  and 

To simplify a fraction you need to find all the factors that the numerator and denominator have in common. You can see that both share 2 so when you divide both by 2 you get

this is close to the answer but it asks for no common factors, it is hard to see but both of these share 7 as a common factor. When you divide both by 7 you get

Finding the least common denominator in rational expressions follows the same procedue as finding the least common denominator in fractions.

The least common denominator for this rational exresspion will use all terms with the highest exponents of each.

The first fraction has  as the highest term, the second fraction has  as the highest term, and the third fraction has  as the highest term. Now we combine these and get the least common denominator to be:

To simplify a fraction, find the gcf (gcd) of the numerator and denominator and divide both by the gcf (gcd). the gcf of  and  is  so:

The first and easiest simplification for this fraction is to divide numerator and denominator by .  This gives you:

Next, notice that the two fractions do not both contain factors of .  This is because the denominator's digits, when added up come to , which is not divisible by .  This means  is not divisible by .  Now,  is .  There is no shared  between these numbers.  However, if you try, you will see that they are both divisible by , which gives you:

This is simplest form.