A rational expression is a fraction whose numerator and/or denominator contains a polynomial. For example:
Note: The denominator of a rational expression cannot be zero because division by zero is undefined.
Multiplying and dividing rational expressions follows rules similar to those for fractions.
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.
Example: (2x / 3y) × (4y2 / 5x)
- 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.
Example: (3x2 / 4y) ÷ (6x / 8y2)
- 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:
4y * 6x = 24xy - 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.
Example: (6x2 - 18x) / 3x
- Factorize the numerator: 6x2 - 18x = 6x(x - 3)
- Factorize the denominator: 3x
- Simplify the resulting expression: (6x(x - 3)) / 3x = 2(x - 3)
Solved Examples
Problem 1: Multiply and simplify (3x2 / 2y) × (4y / 9x).
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).
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)).
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
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)
