Lesson 8.5: Simplify Complex Rational Expressions

New Concept

A complex rational expression, like $\frac{\frac{1}{x}}{\frac{1}{y}}$, is a fraction containing other fractions. In this lesson, you'll master two powerful methods—division and the LCD approach—to transform these expressions into simpler, single fractions.

What’s next

Let's start with the first method. You'll work through interactive examples showing how to simplify these expressions by rewriting them as division problems.

Property

A complex rational expression is a rational expression in which the numerator or denominator contains a rational expression.
We always exclude values that would make any denominator zero. Examples include:

Property

This method treats the main fraction bar as a division operator. The complex expression is rewritten as a division problem, and then simplified by multiplying the first rational expression by the reciprocal of the second.

HOW TO: Simplify a complex rational expression by writing it as division.

1. Step 1. Simplify the numerator and denominator into single rational expressions.
2. Step 2. Rewrite the complex rational expression as a division problem.
3. Step 3. Divide the expressions by multiplying the first by the reciprocal of the second.

Examples

• To simplify $\frac{\frac{1}{4}+\frac{1}{8}}{\frac{1}{2}}$, first simplify the numerator to $\frac{3}{8}$. Then rewrite as $\frac{3}{8} \div \frac{1}{2}$, which becomes $\frac{3}{8} \cdot \frac{2}{1} = \frac{6}{8} = \frac{3}{4}$.

Property

This method clears all fractions by multiplying the numerator and denominator of the complex expression by the Least Common Denominator (LCD) of all the smaller fractions. Since you are multiplying by $\frac{\text{LCD}}{\text{LCD}}$, which is 1, the value of the expression does not change.

HOW TO: Simplify a complex rational expression by using the LCD.

1. Step 1. Find the LCD of all fractions in the complex rational expression.
2. Step 2. Multiply the numerator and denominator by the LCD.
3. Step 3. Simplify the resulting expression.

Examples

• To simplify $\frac{\frac{1}{4}+\frac{1}{8}}{\frac{1}{2}}$, the LCD of all denominators (4, 8, 2) is 8. Multiply top and bottom by 8: $\frac{8(\frac{1}{4}+\frac{1}{8})}{8(\frac{1}{2})} = \frac{2+1}{4} = \frac{3}{4}$.

Property

Before simplifying a complex rational expression, it is crucial to factor all denominators completely.
Factoring reveals the individual components of each denominator, which is necessary for finding the Least Common Denominator (LCD) and for identifying common factors that can be removed to simplify the final expression.
Be sure to start by factoring all the denominators so you can find the LCD.

Examples

• For $\frac{\frac{5}{m^2-5m+6}}{\frac{2}{m-2}-\frac{1}{m-3}}$, first factor $m^2-5m+6$ into $(m-2)(m-3)$. This shows the LCD for all fractions is simply $(m-2)(m-3)$.

• When simplifying $\frac{\frac{y-3}{y^2-9}}{\frac{4}{y+3}}$ as a division problem, you get $\frac{y-3}{y^2-9} \cdot \frac{y+3}{4}$. Factoring $y^2-9$ into $(y-3)(y+3)$ is required to see the common factors that cancel, leaving $\frac{1}{4}$.