Adding and subtracting rational expressions works just like adding and subtracting numerical fractions. To add fractions, we need to find a common denominator. Let’s look at an example of fraction addition.

We have to rewrite the fractions so they share a common denominator before we are able to add. We must do the same thing when adding or subtracting rational expressions.

The easiest common denominator to use will be the least common denominator, or LCD. The LCD is the smallest multiple that the denominators have in common. To find the LCD of two rational expressions, we factor the expressions and multiply all of the distinct factors. For instance, if the factored denominators were [latex]\left(x+3\right)\left(x+4\right)[/latex] and [latex]\left(x+4\right)\left(x+5\right)[/latex], then the LCD would be [latex]\left(x+3\right)\left(x+4\right)\left(x+5\right)[/latex].

Once we find the LCD, we need to multiply each expression by the form of 1 that will change the denominator to the LCD. We would need to multiply the expression with a denominator of [latex]\left(x+3\right)\left(x+4\right)[/latex] by [latex]\frac{x+5}{x+5}[/latex] and the expression with a denominator of [latex]\left(x+4\right)\left(x+5\right)[/latex] by [latex]\frac{x+3}{x+3}[/latex].

Analysis of the Solution

Multiplying by [latex]\frac{y}{y}[/latex] or [latex]\frac{x}{x}[/latex] does not change the value of the original expression because any number divided by itself is 1, and multiplying an expression by 1 gives the original expression.

Simplify Complex Rational Expressions

A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression [latex]\frac{a}{\frac{1}{b}+c}[/latex] can be simplified by rewriting the numerator as the fraction [latex]\frac{a}{1}[/latex] and combining the expressions in the denominator as [latex]\frac{1+bc}{b}[/latex]. We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get [latex]\frac{a}{1}\cdot \frac{b}{1+bc}[/latex], which is equal to [latex]\frac{ab}{1+bc}[/latex].