Multiplication of algebraic fractions is not so difficult. You need to multiply the numerators, denominators of all fractions together to get its product. Before multiplying the algebraic fractions, factorize them. Check out the simple steps of multiplication of algebraic fractions and solved examples in the following sections.
How to find the Product of Algebraic Fractions?
Here we are giving the simple steps to calculate the multiplication of two or more algebraic fractions. Follow these instructions to get the product quickly.
- Find the factors of numerators, denominators of algebraic fractions.
- If there are any like factors, then cancel them.
- Multiply the numerators of remaining factors and denominators.
Solved Examples on Multiplication of Algebraic Fractions
Example 1.
Simplify 5 / (a + a²) x (a³ – a) / ab?
Solution:
Given that,
5 / (a + a²) x (a³ – a) / ab
Get the factors of both fractions
= 5 / a (a + 1) x a (a² – 1) / ab
= 5 / a (a + 1) x [a (a + 1) (a – 1)] / ab
Multiply the two algebraic fractions.
= 5a (a + 1) (a – 1) / a (a + 1) ab
Cancel the terms a (a +1) in both denominator and numerator.
= 5 (a – 1) / ab
Example 2.
Find the product of the algebraic fraction [5a / 2a-1 – (a-2) / a] x [2a / (a+2) – 1 / (a+2)?
Solution:
Given that,
[5a / 2a-1 – (a-2) / a] x [2a / (a+2) – 1 / (a+2)]
The least common multiple of denominators of the first part is a(2a – 1) and the L.C.M of denominators of the second part is a + 2.
Therefore, [5a. a / a(2a – 1) – (a – 2) . (2a – 1) / a(2a – 1)] x [2a / (a+2) – 1 / (a+2)]
= [(5a² – (a – 2) (2a – 1)) / a(2a – 1)] x [(2a – 1) / (a+2)]
= [(5a² – 2a² + a + 4a – 2) / a(2a – 1)] x [(2a – 1) / (a+2)]
= [(3a² + 5a – 2) / a(2a – 1)] x [(2a – 1) / (a+2)]
= [(3a² + 6a – a – 2) / a(2a – 1)] x [(2a – 1) / (a+2)]
= [(3a (a + 2) -1(a + 2)) / a(2a – 1)] x [(2a – 1) / (a+2)]
= [(a + 2) (3a – 1) / a(2a – 1)] x [(2a – 1) / (a+2)]
= [(a + 2) (3a – 1) (2a – 1)] / [a(2a – 1) (a+2)]
Here the common factors in the numerator and denominator are (a+2), (2a-1). Cancel these factors in both to find the lowest form
= (3a – 1) / a
Therefore, [5a / 2a-1 – (a-2) / a] x [2a / (a+2) – 1 / (a+2)] = (3a – 1) / a.
Example 3.
Find the product and express in the lowest form: 5x² / (x² – 2x) x (x² – 4) / (x² + 2x)?
Solution:
Given that,
5x² / (x² – 2x) x (x² – 4) / (x² + 2x)
Ge the factors of both fractions.
= 5x² / x(x – 2) x (x² – 2²) / (x(x + 2))
Cancel the common term x in the first part.
= 5x / (x – 2) x (x + 2) ( x – 2) / (x(x + 2))
Cancel the common factor (x+2) in the second part.
= 5x / (x – 2) x (x – 2) / x
Multiply both numerators and denominators
= 5x . (x – 2) / x . (x – 2)
Cancel the common factor (x – 2) in both numerator and denominator.
= 5x/ x
= 5.
∴ 5x² / (x² – 2x) x (x² – 4) / (x² + 2x) = 5.
Example 4.
Find the product of the algebraic fractions in the lowest form:
[(x + 2y) / (2x + y)] x [(2x + 5y) / (x + y)]
Solution:
Given that,
[(x + 2y) / (2x + y)] x [(2x + 5y) / (x + y)]
= [(x + 2y) (2x + 5y)] / [(2x + y) (x + y)]
= [2x² + 5xy + 4xy + 10y²] / [2x² + xy + 2xy + y²]
= [2x² + 9xy + 10y²] / [2x² + 3xy + y²]
Example 5.
Simplify (4x² – 1) / (9x – 6) x (15x – 10) / (x + 4)?
Solution:
Given that,
(4x² – 1) / (9x – 6) x (15x – 10) / (x + 4)
Calculate the factors.
= ((2x)² – 1²) / 3(3x – 2) x 5(3x – 2) / (x + 4)
= (2x + 1) (2x – 1) / 3(3x – 2) x 5(3x – 2) / (x + 4)
Multiply numerators, denominators together.
= [(2x + 1) (2x – 1) . 5(3x – 2)] / [3(3x – 2) (x + 4)]
Cancel the terms (3x – 2)
= [5(2x + 1) (2x – 1)] / 3(x + 4)
∴ (4x² – 1) / (9x – 6) x (15x – 10) / (x + 4) = [5(2x + 1) (2x – 1)] / 3(x + 4).