In order to find the least common denominator, one of the ways to find it is to write out the factors for each number.  We are trying to find the least number possible that is divisible by all the numbers of the denominators.

Write out the factors for all the numbers.  Notice that the number must be at least greater than \(\displaystyle 20\) because of the four and five in the second and third fractions.

\(\displaystyle 3-[3,6,9,12,15,18...42,45,48,51,54,57,60]\)

\(\displaystyle 4-[4,8,12, 16,20...,40,44,48,52,56,60]\)

\(\displaystyle 5-[5,10,15,20,25,30,35,40,45,50,55,60]\)

The number \(\displaystyle 60\) is divisible by all three numbers.

The answer is:  \(\displaystyle 60\)

The least common denominator is the lowest number that is divisible by all the denominators in the expression.  Write out some of the factors of each denominator.

\(\displaystyle 4-4,8,12,16,18,20\)

\(\displaystyle 10-10,20\)

\(\displaystyle 20-20\)

Notice that each factor shares twenty as the common denominator and can be converted to this denominator to solve.

The answer is:  \(\displaystyle 20\)

In order to find the least common denominator, write out the factors for both denominators.  The LCD is the smallest number divisible by both denominators.

\(\displaystyle 9-[9,18,27,36,45,54,63,72,81,90,99]\)

\(\displaystyle 33-[33,66,99]\)

Notice that they both share 99 as a common denominator.

The answer is:  \(\displaystyle 99\)

The least common denominator is the term that is divisible by both of the uncommon denominators.  

Notice that if we multiplied the \(\displaystyle (x-2)\) quantity by two, we will get the denominator of the second term and the second fraction can be left unchanged.

We do not need to use the FOIL method to determine the least common denominator.

Do not solve the expression.

The answer is:  \(\displaystyle 2x-4\)

The least common denominator can be determined by writing the factors of each denominator provided.  

\(\displaystyle 16-[16,32,48,64,80,96]\)

\(\displaystyle 6-[6,12,18,24,30,36,42,48,...60,66,72,78,84,90,96]\)

\(\displaystyle 8-[8,16,24,32,40,48,...,80,88,96]\)

Notice that the least common denominator is the least number that is divisible by all the denominators of the fractions.  The common factors are 48 and 96.

The first number to appear is 48.

The answer is:  \(\displaystyle 48\)

The least common denominator is the minimal term that is divisible by all the denominators in the fractions.

Notice that the first two fractions has an x-term, and the ten does not have an x-term.  We will need to multiply an x-variable to the ten.

\(\displaystyle 10x\)

Now that all the denominators share an x-term, we will concentrate on the multiples of each denominator coefficient.

\(\displaystyle 4-[4,8,12,16,20]\)

\(\displaystyle 5-[5,10,15,20]\)

\(\displaystyle 10-[10,20]\)

The coefficient must be \(\displaystyle 20\).

This means that the least common denominator must be \(\displaystyle 20x\), since this is divisible by all the denominators in the expression.

The answer is:   \(\displaystyle 20x\)

In order to find the least common denominator, we will need to determine the minimal value that is divisible by all three denominators.

Write out the multiples for every denominator.  Notice that 16 is the minimum that the number that should be started with.

\(\displaystyle 3-[18,21,24,27,30,33,36,39,42,45,48]\)

\(\displaystyle 8-[16,24,32,48]\)

\(\displaystyle 16-[16,32,48]\)

All three denominators share 48 as a result.

The answer is:  \(\displaystyle 48\)

In order to determine the least common denominator, we will need to start with the largest denominator, write the multiples for that denominator, and visually determine whether if the multiples are divisible by the next highest denominator.

\(\displaystyle 32-[32,64,96,128,160,...]\)

Stop at 160, since this is the first multiple that is divisible by both four and twenty.

The means that the least common denominator is 160.

The answer is:  \(\displaystyle 160\)

To simplify the sum of the two fractions, we must find the common denominator.

Simplifying the denominator of the first fraction, we get

\(\displaystyle \frac{3}{x^2-1}=\frac{3}{(x-1)(x+1)}\)

because the denominator is a difference of two squares, which follows the form

\(\displaystyle (x^2-a^2)=(x+a)(x-a)\)

Now, we can rewrite the sum as

\(\displaystyle \frac{3}{(x+1)(x-1)}+\frac{4}{x+1}\)

It is far easier to see the common denominator now:

\(\displaystyle \frac{3}{(x-1)(x+1)}+\frac{4(x-1)}{(x-1)(x+1)}=\frac{3+4x-4}{(x-1)(x+1)}=\frac{4x-1}{x^2-1}\)

Find the least common denominator (LCD) and convert each fraction to the LCD and then add.  Simplify as necessary.

\(\displaystyle \frac{2}{3}+\frac{1}{4}+\frac{5}{6}=\frac{8}{12}+\frac{3}{12}+\frac{10}{12}=\frac{21}{12}\)

The result is an improper fraction because the numerator is larger than the denominator and can be simplified and converted to a mix numeral.

\(\displaystyle \frac{21}{12}=\frac{7}{4}=1\frac{3}{4}\)