Multiply and Divide Rational Expressions

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

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

We can perform various operations on rational expressions just like fractions. So, to multiply and divide rational expressions, we apply the same rules for multiplying and dividing fractions as we do with numbers.

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

Steps to Multiply and Divide Rational Expressions

Let's discuss the 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:
• 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)

Related Articles

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

Solved Problems of 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)