Multiply and Divide Rational Expressions - GeeksforGeeks (2024)

Rational expressions, much like fractions, can be multiplied and divided using a systematic approach that simplifies the process while maintaining accuracy. In mathematics, mastering the multiplication and division of rational expressions is crucial, especially in algebra, where these operations are frequently encountered. By learning how to handle these expressions efficiently, you can solve complex equations and simplify problems that involve variables and fractions.

This article will guide you through the steps to multiply and divide rational expressions with ease, ensuring a solid foundation in this essential algebraic skill.

What are Rational Expressions?

Rational expressions are mathematical expressions that represent the ratio of two polynomials. In other words, a rational expression is a fraction where both the numerator and the denominator are polynomials.

Definition of a Rational Expressions

A rational expression is a fraction in which both the numerator and the denominator are polynomials.

For example: (3x² + 2x – 5) / (x² – 4).

Note: If a represents any number, then a ÷ 0 is considered Undefined.

Steps to Multiply and Divide Rational Expressions

Let’s discuss multiplication and division of rational expressions separately as follows:

Multiplying Rational Expressions

To multiply two rational expressions, follow these steps:

Step 1. Factorize the numerators and denominators, if possible.

Step 2. Multiply the numerators together.

Step 3. Multiply the denominators together.

Step 4. Simplify the resulting expression by canceling out common factors.

Let’s consider an example for better understanding.

Example: (2x / 3y) × (4y² / 5x)

Solution:

• Factorize (if necessary): 2x and 4y² are already in simplest form.
• Multiply the numerators: 2x × 4y² = 8xy²
• Multiply the denominators: 3y × 5x = 15xy
• Simplify the resulting expression: 8xy² / 15xy = 8y / 15

Dividing Rational Expressions

To divide one rational expression by another, follow these steps:

Step 1. Factorize the numerators and denominators, if possible.

Step 2.Take the reciprocal of the divisor.

Step 3. Multiply the first rational expression by the reciprocal of the second.

Step 4. Simplify the resulting expression by canceling out common factors.

Let’s consider an example for better understanding.

Example: (3x² / 4y) ÷ (6x / 8y²)

Solution:

• Factorize (if necessary): 3x² and 6x are already in simplest form. 4y and 8y² can be factorized as (2² × y) and 2³ × y².
• Take the reciprocal of the divisor: (3x² / 4y) × (8y² / 6x)
• Multiply the numerators: 3x² × 8y² = 24x²y²
• Multiply the denominators: [Tex]4y * 6x = 24xy[/Tex]
• Simplify the resulting expression: (24x²y² )/( 24xy) = xy

Simplifying Rational Expressions

Simplification involves reducing the rational expression to its lowest terms. This requires factoring both the numerator and the denominator and canceling out common factors.

Let’s consider an example for better understanding.

Example: (6x² – 18x) / 3x

Step-by-step solution:

• Factorize the numerator: 6x² – 18x = 6x(x – 3)
• Factorize the denominator: 3x
• Simplify the resulting expression: (6x(x – 3)) / 3x = 2(x – 3)

Conclusion

This article is intended to offer a thorough manual on the multiplication and division of rational expressions. Upon completion of this article, students will possess the skills to effectively manage these expressions, streamline them, and utilize the concepts to tackle problems confidently. The article will encompass essential topics and subtopics, incorporate pertinent keywords, and present examples and FAQs to guarantee a comprehensive grasp of the concepts.

Related Articles

• Rational Expression
• Rational Number
• Multiplication of Rational Number
• Division of Rational Number

Solved Problems: Multiply and Divide Rational Expressions

Problem 1: Multiply and simplify (3x² / 2y) × (4y / 9x).

Solution:

Multiply the numerators: 3x² × 4y = 12x²y

Multiply the denominators: 2y × 9x = 18xy

Simplify the resulting expression: 12x²y / 18xy = 2x / 3.

Problem 2: Divide and Simplify (5x / 6y) ÷ (10x² / 12y²).

Solution:

Take the reciprocal of the divisor: (5x / 6y) × (12y² / 10x²)

Multiply the numerators: 5x × 12y² = 60xy²

Multiply the denominators: 6y × 10x²= 60yx²

Simplify the resulting expression: 60xy² / 60yx² = y / x.

Problem 3: Simplify ((x² – 4) / (x² + 4x + 4)) × ((x + 2) / (x – 2)).

Solution:

Factorize the numerator and the denominator where possible:

Numerator:

(x²– 4) = (x – 2)(x + 2)

Denominator:

(x² + 4x + 4) = (x + 2)(x + 2)

Simplify by canceling out common factors:

((x – 2)(x + 2) / (x + 2)(x + 2) )* (x + 2) / (x – 2) = 1 .

Practice Problems: Multiply and Divide Rational Expressions

Problem 1: Multiply and simplify: (4x/5y) × (10y²/8x)

Problem 2: Divide and simplify: (7x³/9y) / (14x²/27y²)

Problem 3: Simplify: ((2x²-8)/4x) × (6x/(x-2))

Problem 4: Multiply and simplify: ((3a²-9a)/2b) × (4b/6a)

Problem 5: Divide and simplify: (5m/6n²) / (10m²/12n)

FAQs:

What is a rational expression?

A rational expression is a fraction in which the numerator and denominator are both polynomials. For instance, (x^2 – 1) / (x + 2) is a rational expression.

How can rational expressions be multiplied?

Rational expressions can be multiplied by multiplying the numerators together and the denominators together, followed by simplifying the result through canceling out common factors.

How can rational expressions be divided?

Rational expressions can be divided by multiplying the first expression by the reciprocal of the second expression, and then simplifying the result by canceling out common factors.

Can you give an example of multiplying rational expressions?

For example,[Tex] (2x / 3y) \times (4y^2 / 5x) = 8y / 15.[/Tex]

Can you provide an example of dividing rational expressions?

For example, [Tex](3x^2 / 4y) ÷ (6x / 8y^2) = xy.[/Tex]

Why is it important to factorize rational expressions?

The factorization of rational expressions is crucial because it simplifies the process by allowing the cancellation of common factors in both the numerator and denominator.

What should I do if the denominator of a rational expression is zero?

In case the denominator of a rational expression equals zero, the expression becomes undefined. It is essential to verify and eliminate such values from the domain.