Simplify the rational expressions below: 6 x 12 x \frac{6x}{12x}

In recent videos, we got introduced to rational expressions. We talked about what these are and how they are defined. Now in this video, we're going to learn about how we can simplify rational expressions. And I think you're gonna find this process is actually relatively straightforward to what we already know about simplifying rational numbers. So let's just go ahead and get right into things. Now, first off, I want you to recall how we could simplify a rational number, which is also just simplifying a fraction. What you first wanna do is take your fraction and factor it completely. So factoring the top 28, this will give us seven times four. And factoring the bottom 35, we get seven times five. Notice how seven seven is a common factor, which we can cancel, giving us the simplified answer of four over five. Now, it turns out simplifying rational expressions is actually very similar. What you want to do is factor the numerator and denominator completely, and from there, you're going to cancel the common factors. So let's go ahead and try simplifying this rational expression. Now 28, I know that's the same thing as seven times four. We actually factored 28 over here, but seven times four is broken down into prime numbers. And then for this x cubed, well, that's just the same thing as x times x times x. So, this is what we get on top. Now, on bottom, in the denominator, we have 35. 35 breaks down into seven times five, and then notice we have x to the fifth, which is the same thing as x times x times x times x times x. So, now we have this fully factored. Now, from here, what you want to do is cancel the common factors. Now, I can see we have a seven in common in the numerator and denominator. So what I can do is cancel the sevens right here. Now, notice also I have three x's in the numerator and three x's in the denominator, which are going to cancel each other completely. So what that means is in the numerator, we'll have four left over, and then in the denominator, we have a five left over, as well as two more x's, which I can also just write as x squared. So four over five x squared is the simplified answer to this rational expression. So this is how you can simplify rational expressions. Now I wanna make sure we can do this in an example as well, so let's try a couple more examples here to really make sure we have this down. Now this problem asks us to simplify the following expressions, and we'll start with example a. Notice here that everything is already factored. We have x plus two times x minus two on top, with an x plus six times x plus two on bottom. Since this is already factored, we can just cancel the common factors, which I can see would be x plus two. So that means in the numerator, we'll only have x minus two left over, and then in the denominator, we'll only have x plus six left over. And this would be our simplified answer, and that's it. So as you can see, when things are already factored, it's very straightforward. But now let's take a look at example b. In this example, notice that things are not fully factored here. I mean, we have the x minus five, which is fully factored. We have a three x squared minus 15 x, which is not. But that's okay because we can factor this completely. We'll keep the x minus five on top, and then on the bottom, I'm going to pull out the greatest common factor, which is three x. If I pull a three x to the outside here, inside parentheses, we'll have three x times whatever would give us x squared, which I know is just x, and that's going to be minus, and that's going to be three x times whatever would give us 15 x, which is five. Notice if I redistributed this three x into the parentheses, I would get the same thing that we started with. So we did this correctly. Now the reason that I factored it like so is notice we now have the common factor x minus five, which is going to cancel. So all we're gonna end up getting here is one over three x, and that right there is the simplified answer and the solution to this problem. So this is how you can simplify rational expressions. Hope you found this video helpful, and definitely check out the practice and examples after this video to make sure we have this concept down. See you all in the next one.

Table of contents

Skip topic navigation

1. Review of Real Numbers2h 43m

Simplifying Fractions
30m

Evaluating Exponents
29m

Evaluating Expressions
15m

Translating Phrases to Expressions
16m

Translating Sentences to Equations
9m

The Distributive Property
17m

Simplifying Expressions
44m

2. Linear Equations and Inequalities5h 35m

Solving Linear Equations
2h 23m

Linear Inequalities in One Variable
46m

Set Operations and Compound Inequalities
1h 7m

Absolute Value Equations
38m

Absolute Value Inequalities
39m

3. Solving Word Problems2h 46m

Introduction to Problem Solving
37m

Formulas
25m

Percent Problem Solving
59m

Mixture Problem Solving
43m

4. Graphs and Functions5h 12m

The Rectangular Coordinate System
44m

Graph Linear Equations in Two Variables
24m

Graph Linear Equations Using Intercepts
23m

Slope of a Line
44m

Slope-Intercept Form
38m

Point Slope Form
22m

Linear Inequalities in Two Variables
28m

Introduction to Relations and Functions
53m

Function Notation
15m

Composition of Functions
17m

5. Systems of Linear Equations1h 53m

Solving Systems of Linear Equations by Graphing
29m

Solving Systems of Linear Equations by Substitution
15m

Solving Systems of Linear Equations by Elimination
45m

Systems of Linear Inequalities
23m

6. Exponents, Polynomials, and Polynomial Functions3h 17m

The Product Rule
9m

The Quotient Rule
10m

Intro to the Power Rules
17m

The Power of a Quotient Rule
13m

Negative Exponents
24m

Simplifying Exponential Expressions Using All Exponent Rules
6m

Intro to Polynomials
21m

Adding and Subtracting Polynomials
20m

Multiplying Polynomials
38m

Special Products
34m

7. Factoring2h 49m

Factoring by Greatest Common Factor & Grouping
37m

Factoring Trinomials of the Form x² + bx + c
11m

Factoring Trinomials of the Form ax² + bx + c
37m

Factoring Special Products
41m

Quadratic Equations & Applications
41m

8. Rational Expressions and Functions3h 44m

Simplifying Rational Expressions
42m

Multiplying and Dividing Rational Expressions
25m

Adding and Subtracting Rational Expressions with Common Denominators
19m

Least Common Denominators
32m

Adding and Subtracting Rational Expressions with Different Denominators
32m

Rational Equations
44m

Direct & Inverse Variation
27m

9. Roots, Radicals, and Complex Numbers3h 56m

Introduction to Radical Expressions
46m

Simplifying Radical Expressions
47m

Adding and Subtracting Radicals
19m

Multiplying, Dividing, and Rationalizing Radicals
53m

Introduction to Complex Numbers
18m

Multiplying and Dividing Complex Numbers
51m

10. Quadratic Equations and Functions3h 1m

The Square Root Property
25m

Completing the Square
26m

The Quadratic Formula
40m

Graphing Quadratic Equations
1h 28m

11. Inverse, Exponential, & Logarithmic Functions2h 31m

Introduction to Inverse Functions
25m

Exponential Functions
35m

Introduction to Logarithmic Functions
33m

Properties of Logarithms
55m

12. Conic Sections & Systems of Nonlinear Equations58m

Graphing Circles
32m

Ellipses
15m

Parabolas
10m

13. Sequences, Series, and the Binomial Theorem1h 51m

Introduction to Sequences
40m

Arithmetic Sequences
27m

Geometric Sequences
29m

Factorials
13m

8. Rational Expressions and Functions

Simplifying Rational Expressions

Intermediate Algebra

8. Rational Expressions and Functions

Simplifying Rational Expressions

Previous problemNext problem

Struggling with Intermediate Algebra?

Join thousands of students who trust us to help them ace their exams!

Multiple Choice

Simplify the rational expressions below:
$6 x over 12 x$

$2x$

$2$

$$\frac$12x$

$$\frac$12$

0 Comments

Previous problemNext problem

Verified step by step guidance

Start with the given rational expression: $\frac{6x}{12x}$.

Identify common factors in the numerator and denominator. Both numerator and denominator have a factor of \$6$ and a variable $x$.

Factor out the common terms: $\frac{6 \times x}{12 \times x}$ can be rewritten as $\frac{6}{12} \times \frac{x}{x}$.

Simplify the fraction $\frac{6}{12}$ by dividing numerator and denominator by their greatest common divisor, which is 6.

Since $\frac{x}{x} = 1$ (for $x \neq 0$), multiply the simplified fraction by 1 to get the final simplified expression.

5:34m

Watch next

Master Simplifying Rational Expressions Example 1 with a bite sized video explanation from Patrick Ford

Struggling with Intermediate Algebra?

Simplify the rational expressions below:6x12x\(\frac{6x}{12x}\)

Watch next

Simplify the rational expressions below:
$6 x over 12 x$