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Dividing Rational Expressions

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Dividing Rational Expressions

Study Guide

Key Definition

To divide a rational expression by another, multiply by the reciprocal: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Important Notes

Always check the denominators are not zero $b \neq 0, c \neq 0, d \neq 0$
Multiply by the reciprocal of the second expression
Simplify by factoring and canceling common terms
Ensure the final result is in its simplest form
Remember that dividing by a fraction is the same as multiplying by its reciprocal

Mathematical Notation

$\frac{a}{b}$ represents a fraction

$\div$ represents division

$\times$ represents multiplication

$x^2$ represents x squared

$\sqrt{x}$ represents the square root of x

Remember to use proper notation when solving problems

Why It Works

Every nonzero number has a reciprocal that yields 1 when multiplied. Dividing by $\frac{c}{d}$ is the same as multiplying by its reciprocal $\frac{d}{c}$, which preserves the value of the expression. This conversion turns division into multiplication, making it easy to cancel common factors and simplify.

Remember

Always factor expressions and cancel out common terms to simplify: $\frac{(x+2)(x-2)}{(x+6)(x+2)}$

Quick Reference

Division by Fraction:$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Understanding Dividing Rational Expressions

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

First, rewrite the division as multiplication by the reciprocal: (a/b) ÷ (c/d) becomes (a/b) × (d/c). Then multiply across the numerators and denominators directly.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is $\frac{x^2 - 4}{x+6} \div \frac{x+2}{2(x+6)}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboarder divides the length of a ramp $\frac{2x+3}{x-1}$ by its height $\frac{x+2}{x-1}$. What is the simplified expression for the ramp's slope?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

How can you simplify $\frac{x^2+5x+6}{x^2-4} \div \frac{x+3}{x-2}$?

Challenge Quiz

Single Choice Quiz

Advanced

Simplify $\frac{x^3 - x}{x^2 - 1} \div \frac{x}{x+1}$.

Recap

Watch & Learn

Review key concepts and takeaways