Simplifying a Complex Rational Expression

(Videos) Complex Fractions

by mathman1024

Here's a video by mathman1024 showing you how to simplify complex fractions. Complex fractions are fractions where there's a fraction in the numerator and denominator.

The example he looks at first is $\dfrac{\frac75+4}{7-\frac7{10}}$. One way to solve these problems is to turn the numerator into a single fraction and turn the denominator into a single fraction.

Let's start by just looking at the numerator. $$\begin{align} &\frac75+4\\ &=\frac75+4\color{#8925AE}{\cdot\frac55}\\ &=\frac75+\frac{20}5\\ &=\frac{27}5 \end{align}$$ Next let's do the same thing to the denominator. $$\begin{align} &7-\frac7{10}\\ &=7\color{#8925AE}{\cdot\frac{10}{10}}-\frac{7}{10}\\ &=\frac{70}{10}-\frac7{10}\\ &=\frac{63}{10} \end{align}$$ So now our problem becomes $$\begin{align} \dfrac{\frac75+4}{7-\frac7{10}}&=\dfrac{\frac{27}5}{\frac{63}{10}}\\ &=\frac{27}5\cdot\frac{10}{63}\\ &=\frac{27}{63}\cdot\frac{10}5\\ &=\frac{\color{#8925AE}{9}\cdot3}{\color{#8925AE}{9}\cdot7}\cdot\frac{\color{#8925AE}{5}\cdot2}{\color{#8925AE}{5}\cdot1}\\ &=\frac67 \end{align}$$

He then goes over a different way to solves these problems. Both give the same answer and both are fine. Whatever you think is easiest is what you should use.

In the last method we focused on the numerator and denominator separately. In this method, we focus on the entire complex fraction. So we begin by finding the LCD of the entire fraction. We want this to clear out the fractions in the numerator and denominator. Here, our LCD is $10$. $$\begin{align} &\dfrac{\frac75+4}{7-\frac7{10}}\\ &=\color{#8925AE}{\frac{10}{10}}\cdot\dfrac{\frac75+4}{7-\frac7{10}}\\ &=\dfrac{14+40}{70-7}\\ &=\dfrac{54}{63}\\ &=\frac67 \end{align}$$

(Video) Complex Fraction Example

by mathman1024

Here's another video by mathman1024 with some more examples. Try these ones on your own and then watch the video to see the answers!

1. $\dfrac{4-\frac47}{\frac13-\frac17}$
2. $\dfrac{x+\frac8{x^2}}{x-\frac4x}$

Background

Simplifying complex rational expressions takes a little more work than simpler rational expressions, but you will use the same techniques that you learned when adding/subtracting and multiplying/dividing rational expressions.

Simplifying Complex Rational Expressions

Complex rational expressions tend to have fractions within fractions, which make them complex. The key to simplifying complex rational expressions is to combine all fractions into one fractional expression.

$$\frac{ \frac{x+1}{x+2} +3}{ \frac{2x+4}{x+2}}$$

In this above example, we can see that the numerator has a fraction within itself. There is also a fraction in the denominator. Our first step is to reduce the fractions in the numerator and denominator. There are a variety of tricks you will use from previous math lessons that will help you here. Some potential tricks to keep in mind: rationalize or multiply the numerator, find a common denominator by the LCD, and combining like terms. If you're dividing radical expressions, you may need to rationalize the denominator.

Let's start with the numerator.

$$\begin{align} \frac{ \frac{x+1}{x+2} +3}{ \frac{2x+4}{x+2}} \\ &=\frac{ \frac{x+1}{x+2} +\frac{(3)(x+2)}{x+2}}{ \frac{2x+4}{x+2}} \\ &=\frac{ \frac{x+1}{x+2} +\frac{3x+6}{x+2}}{ \frac{2x+4}{x+2}} \\ &=\frac{ \frac{x+1+3x+6}{x+2}}{ \frac{2x+4}{x+2}} \\ &=\frac{ \frac{4x+7}{x+2}}{ \frac{2x+4}{x+2}} \\ \end{align}$$

We've reduced the numerator! Taking a look at the denominator, we can see that it doesn't need to be reduced. Let's rewrite our expression into an easier format to understand. Then, we'll simplify! Remember

$$\begin{align} \frac{4x+7}{x+2} \div \frac{2x+4}{x+2} \\ &=\frac{4x+7}{x+2} \cdot \frac{x+2}{2x+4} \\ &=\frac{4x+7}{1} \cdot \frac{1}{2x+4} \\ &=\frac{4x+7}{2x+4} \\ \end{align}$$

Simplifying Complex Rational Expressions

A complex rational expression is a big fraction that has smaller fractions in the numerator, denominator, or both. They often look really scary when you first look at them - it's a fraction inside of a fraction! These can be tricky, but we can simplify them to make them easier to work with.

In this lesson, you will learn three steps to tackle this inception-style fraction.

There are three main steps to simplifying a complex rational expression.

Step 1: Find the least common denominator (LCD) of all the smaller fractions in the expression.

Step 2: Multiply the numerator and denominator of the complex fraction by the LCD to eliminate the smaller fractions.

Step 3: Simplify the resulting expression by factoring and canceling any common factors.