## Simplifying 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 $\frac{a}{\frac{1}{b}+c}$ can be simplified by rewriting the numerator as the fraction $\frac{a}{1}$ and combining the expressions in the denominator as $\frac{1+bc}{b}$. We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get $\frac{a}{1}\cdot \frac{b}{1+bc}$, which is equal to $\frac{ab}{1+bc}$.

### How To: Given a complex rational expression, simplify it.

1. Combine the expressions in the numerator into a single rational expression by adding or subtracting.
2. Combine the expressions in the denominator into a single rational expression by adding or subtracting.
3. Rewrite as the numerator divided by the denominator.
4. Rewrite as multiplication.
5. Multiply.
6. Simplify.

### Example 6: Simplifying Complex Rational Expressions

Simplify: $\frac{y+\frac{1}{x}}{\frac{x}{y}}$ .

### Solution

Begin by combining the expressions in the numerator into one expression.

$\begin{array}{cc}y\cdot \frac{x}{x}+\frac{1}{x}\hfill & \text{Multiply by }\frac{x}{x}\text{to get LCD as denominator}.\hfill \\ \frac{xy}{x}+\frac{1}{x}\hfill & \\ \frac{xy+1}{x}\hfill & \text{Add numerators}.\hfill \end{array}$

Now the numerator is a single rational expression and the denominator is a single rational expression.

$\frac{\frac{xy+1}{x}}{\frac{x}{y}}$

We can rewrite this as division, and then multiplication.

$\begin{array}{cc}\frac{xy+1}{x}\div \frac{x}{y}\hfill & \\ \frac{xy+1}{x}\cdot \frac{y}{x}\hfill & \text{Rewrite as multiplication}\text{.}\hfill \\ \frac{y\left(xy+1\right)}{{x}^{2}}\hfill & \text{Multiply}\text{.}\hfill \end{array}$

### Try It 5

Simplify: $\frac{\frac{x}{y}-\frac{y}{x}}{y}$

Solution

### Can a complex rational expression always be simplified?

Yes. We can always rewrite a complex rational expression as a simplified rational expression.