Just as with fractions, when we add or subtract rational expression, we must have a common denominator.

When we deal with rational expressions with monomial denominators, we find the common denominator for the coefficients, and for variables we use the highest exponent.

Here is an example:

In this problem we have monomial denominators, so we look at the coefficients, 6 and 8, a common denominator for these is 24. We then look at each variable individually. In this problem we have x³ and x², since 3 is our largest exponent we will use x³ in our LCD. We also have y⁵ and y⁷, since 7 is our larger exponent we will use y⁷ in the LCD.

So our LCD in this problem would be 24x³y⁷

If our rational expressions have polynomial denominators, then to find the LCD we must first factor each denominator. After we factor the denominators, we get the LCD by writing each factor only once. The only time a factor will appear twice in the LCD is if it appears twice in a single denominator.

Here is an example:

We first factor our denominators:

Now we write down each factor once to get our LCD. LCD = (x + 2)(x + 6)(x - 3). Note we do not write (x + 6) twice because it does not appear in a single denominator twice. The only time a factor will appear twice in the LCD is if it appears twice in a single denominator.

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#1 solution

In this problem we have monomial denominators, so we look at the coefficients, 4 and 10, a common denominator for these is 20. We then look at each variable individually. In this problem we have x⁴ and x², since 4 is our largest exponent we will use x⁴ in our LCD. We also have y³ and y⁵, since 5 is our larger exponent we will use y⁵ in the LCD.

So our LCD in this problem would be 20x⁴y⁵

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#2 solution

Since we have polynomial denominators, we first factor our denominators:

Now we write down each factor once to get our LCD. LCD = (2x + 1)(x - 5)(2x + 1), or (2x + 1)²(x - 5).

Note we write (2x + 1) twice because it appears in a single denominator twice. The only time a factor will appear twice in the LCD is if it appears twice in a single denominator.

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#3 solution     

We first factor our denominators:

Now we write down each factor once to get our LCD. LCD = (x + 9)(x - 9)(x + 2).

Note we do not write (x - 9) twice because it does not appear in a single denominator twice. The only time a factor will appear twice in the LCD is if it appears twice in a single denominator.