How to simplify rational expressions? Grade 11 examples are presented along with detailed solutions and full explanations. More questions with answers at the bottom of the page are also included.
An online calculator to simplify rational expressions is included and may be used to check results.

How to simplify rational expressions?

How to simplify rational expressions? The rules of addition, subtraction, multiplication and division of rational expressions are used to simplify complex expressions.
Rational expressions with the same denominator are added or subtracted as follows:

If you have difficulties in simplifying rational expressions, review the tutorials in How to Add, Subtract and Simplify Rational Expressions and How to multiply, divide and simplify rational expressions and then start the present tutorial. The examples with detailed solutions and explanations in this tutorials will help you overcome any difficulties in simplifying rational expressions on the condition that you understand every step involved in solving these questions and also spend more time practicing if needed. I will present the examples below starting with fractions first and then with rational expressions, with more challenging questions as you walk through the tutorial. You need to understand each step!
You may also want to review the questions on reducing rational expressions and their solutions

Example 1: Simplify: \( \dfrac{\dfrac{2}{3} + 5}{4} +\dfrac{1}{2} \).

Solution:
We first simplify the numerator \( \dfrac{2}{3} + 5 \) of the fraction \( \dfrac{\dfrac{2}{3} + 5}{4} \). Convert to common denominator.

\( \dfrac{\dfrac{2}{3} + 5}{4} +\dfrac{1}{2} = \dfrac{\dfrac{2}{3} + 5 \cdot \dfrac{3}{3}}{4} +\dfrac{1}{2} = \dfrac{\dfrac{17}{3}}{4} +\dfrac{1}{2} \)

Change 4 to a fraction 4 / 1 and simplify \( \dfrac{\dfrac{17}{3}}{4} \) by dividing the fractions which comes to a multiplication by the reciprocal.

\( = \dfrac{\dfrac{17}{3}}{\dfrac{4}{1}} +\dfrac{1}{2} = \dfrac{17}{3} \cdot \dfrac{1}{4} +\dfrac{1}{2} \)

Multiply the fractions \( \dfrac{17}{3} \cdot \dfrac{1}{4} \).

\( = \dfrac{17 \cdot 1}{3 \cdot 4}+\dfrac{1}{2} = \dfrac{17}{12} +\dfrac{1}{2} \)

Convert to common denominator and add.

\( = \dfrac{17}{12} +\dfrac{1}{2} \cdot \dfrac{6}{6} = \dfrac{17+6}{12} = \dfrac{23}{12} \)

Example 5: Simplify: \( \dfrac{ \dfrac{2-x}{x+2} \cdot \dfrac{4}{x+3}}{\dfrac{1}{x+3}-\dfrac{4}{x+3}} \)

Solution:

Apply the rule of multiplication to \( \dfrac{2-x}{x+2} \cdot \dfrac{4}{x+3} \) and the rule of subtraction to \( \dfrac{1}{x+3}-\dfrac{4}{x+3} \).

\( \dfrac{ \dfrac{2-x}{x+2} \cdot \dfrac{4}{x+3}}{\dfrac{1}{x+3}-\dfrac{4}{x+3}} = \dfrac{ \dfrac{4(2-x)}{(x+2)(x+3)}}{\dfrac{1 - 4}{x+3}} \)

Apply the rule of division:

\( = \dfrac{4(2-x)}{(x+2)(x+3)} \cdot \dfrac{x+3}{-3} \)

Simplify.

\( = \dfrac{4(2-x)}{(x+2){\cancel{(x+3)}}} \cdot \dfrac{{\cancel{(x+3)}}}{-3} = -\dfrac{4(2-x)}{3(x+2)} \;\; \text{for} \;\; x \ne -3 \)

Example 6: Simplify: \( \dfrac{1}{1+\dfrac{1}{x+\dfrac{1}{x+1}}} \).

Solution:

We first convert the terms in \( x + \dfrac{1}{x+1} \) to the same denominator.

\( \dfrac{1}{1+\dfrac{1}{x+\dfrac{1}{x+1}}} = \dfrac{1}{1+\dfrac{1}{x \cdot \dfrac{x+1}{x+1}+\dfrac{1}{x+1}}} \)

We now add \( x \cdot \dfrac{x+1}{x+1}+\dfrac{1}{x+1} \) and simplify.

\( = \dfrac{1}{1+\dfrac{1}{\dfrac{x^2+x+1}{x+1}}} \)

Note that \( \dfrac{1}{\dfrac{x^2+x+1}{x+1}} \) is a reciprocal is equal to \( \dfrac{x+1}{x^2+x+1}\). Hence

\(= \dfrac{1}{1+ \dfrac{x+1}{x^2+x+1}} \)

Convert the terms in \( 1+ \dfrac{x+1}{x^2+x+1} \) to the same denominator.

\( = \dfrac{1}{1 \cdot \dfrac{x^2+x+1}{x^2+x+1}+ \dfrac{x+1}{x^2+x+1}} \)

Simplify and apply the rule of addition to \( \dfrac{x^2+x+1}{x^2+x+1}+ \dfrac{x+1}{x^2+x+1} \).

\( = \dfrac{1}{ \dfrac{x^2+x+1}{x^2+x+1}+ \dfrac{x+1}{x^2+x+1}} = \dfrac{1}{ \dfrac{x^2+x+1+x+1}{x^2+x+1}} = \dfrac{1}{ \dfrac{x^2+2x+2}{x^2+x+1}}\)

The last expression is the reciprocal of a rational expression which may be written as.

\( = \dfrac{x^2+x+1}{x^2+2x+2} \)

More Questions: Simplify the following expressions - Answers at the bottom of the page.

a) \( \dfrac{\dfrac{2}{5} + 7}{\dfrac{4}{3}} +\dfrac{1}{3} \)

b) \( \dfrac{\dfrac{x-2}{x+3}+\dfrac{x}{x+3}}{\dfrac{1}{2x+6}} \)

c) \( \dfrac{\dfrac{x-1}{3x-12}+3}{\dfrac{x+1}{x-4}} - \dfrac{2}{3} \)

d) \( \dfrac{ \dfrac{2-x}{x+1}-\dfrac{4}{x+1}}{\dfrac{1}{x-5}-\dfrac{4}{x-5}} \)

e) \( \dfrac{ \dfrac{2-x}{x+2} - \dfrac{4}{x+3}}{\dfrac{1}{x+3} \cdot \dfrac{4}{x+3}} \)

f) \( \dfrac{\dfrac{1}{2}}{1+\dfrac{1}{\dfrac{1}{x-1}-2}} \)

Answers to the above questions

a) \( \dfrac{353}{60} \)

b) \( 4(x-1) \)

c) \( \dfrac{8x-39}{3(x+1)}\)

d)\( \dfrac{ (x+2 ) (x-5)}{3(x+1)} \)

e) \( \dfrac{(-x^2-5x-2)(x+3)}{4(x+2)} \)

f)\( \dfrac{-2x+3}{2(-x+2)} \)

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