Multiplication and division of algebraic fractions

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Multiplication and division of algebraic fractions

Multiplication and division of algebraic fractions

Multiplication of algebraic fractions: In the case of multiplication of monomial algebraic fractions, we should multiply the numerators and denominators separately. Then, the product is reduced to the lowest terms by using the laws of indices. For example,



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Example of multiplication of monomial algebraic fractions

In the case of multiplication of polynomial algebraic fractions, the numerators and denominators are factorized. Then, they are reduced to their lowest terms. For example,        

Example of multiplication of polynomial algebraic fractions

Division of algebraic fractions: In the case of division of algebraic fractions, we should multiply the dividend by the reciprocal of the divisor as like the process of multiplication of algebraic fractions. For example,       

Examples of division of algebraic fractions

Look at the following worked out examples:

Workout Examples

Example 1: Simplify: (i) (3a^2)/(5b^2 ) × (15ab^3)/(27a^2 ) (ii) 2x/(2x+y) × 〖2xy+y〗^2/(8y^3 )

Solution: 1(i) (3a^2)/(5b^2 )× (15ab^3)/(27a^2 ) = (45a^3 b^3)/(135a^2 b^2 ) = ab/3

Solution: 1(ii) 2x/(2x+y) × 〖2xy+y〗^2/(8y^3 ) = 2x/(2x+y) × (y(2x+y))/(8y^3 ) = (2xy(2x+y))/(8y^3 (2x+y)) = x/(4y^2 )


Example 2: Simplify: (i) (a^2+ ab)/(a^2- ab) × (ab-〖 b〗^2)/(a^2-b^2 ) (ii) (x^2-9)/(x^2+4x) × (x^2+ 2x-8)/(x^2+ x - 6)

Solution: 2(i) (a^2+ ab)/(a^2- ab) × (ab-〖 b〗^2)/(a^2-b^2 ) = (a(a+b))/(a(a-b)) × (b(a-b))/((a+b)(a-b)) = b/(a-b)


Solution: 2(ii) (x^2-9)/(x^2+4x) × (x^2+ 2x-8)/(x^2+ x - 6) = (x^2-3^2)/(x(x+4)) × (x^2+ 4x-2x-8)/(x^2+ 3x-2x - 6) = ((x+3)(x-3))/(x(x+4)) × (x(x+4) -2(x+4))/(x(x+3) -2(x+3)) = ((x+3)(x-3))/(x(x+4)) × ((x+4)(x-2))/((x+3)(x-2)) = (x-3)/x

Example 3: Simplify: (3x^2)/(x+2) × 2/(x-2)× (x^2-4)/(15x^2 y)

Solution: (3x^2)/(x+2) × 2/(x-2)× (x^2-4)/(15x^2 y) = (3x^2)/(x+2) × 2/(x-2)× (x^2-2^2)/(15x^2 y) = (3x^2)/(x+2) × 2/(x-2)× ((x+2)(x-2))/(15x^2 y) = 2/5y

Example 4: Simplify: (i) (3a^2 b)/4xy ÷ (6ab^2)/(8xy^2 ) (ii) (x^2-y^2)/xy ÷ (x-y)/y^2

Solution: 4(i) (3a^2 b)/4xy ÷ (6ab^2)/(8xy^2 ) = (3a^2 b)/4xy × (8xy^2)/(6ab^2 ) = (24a^2 bxy^2)/(24xyab^2 ) = ay/b

Solution: 4(ii) (x^2-y^2)/xy ÷ (x-y)/y^2 = (x^2-y^2)/xy × y^2/(x-y) = ((x+y)(x-y))/xy × y^2/(x-y) = (y(x+y))/x


Example 5: Simplify: (i) (3x^2-4x-7)/(x^2-1) ÷ (3x^2-7x)/(x+1) (ii) (2x/(x-1) ÷ (x+1)/1) ÷ 2/(x^2-1)

Solution: 5(i) (3x^2-4x-7)/(x^2-1) ÷ (3x^2-7x)/(x+1) = (3x^2-4x-7)/(x^2-1) × (x+1)/(3x^2-7x) = (3x^2-7x+3x-7)/(x^2-1^2 ) × (x+1)/(x(3x-7)) = (x(3x-7) +1(3x-7))/((x+1)(x-1)) × (x+1)/(x(3x-7)) = ((3x-7)(x+1))/((x+1)(x-1)) × (x+1)/(x(3x-7)) = ((x+1))/(x(x-1))

Solution: 5(ii) (2x/(x-1) ÷ (x+1)/1) ÷ 2/(x^2-1) = (2x/(x-1) × 1/(x+1)) × (x^2-1)/2 = 2x/(x-1) × 1/(x+1) × (x^2-1^2)/2 = 2x/(x-1) × 1/(x+1) × ((x+1)(x-1))/2 = x


Example 6: Simplify: (x^2+3x+2)/(x^2-4x-12) ÷ (x^2-x-6)/(x^2-9x+18)

Solution: (x^2+3x+2)/(x^2-4x-12) ÷ (x^2-x-6)/(x^2-9x+18) = (x^2+3x+2)/(x^2-4x-12) × (x^2-9x+18)/(x^2-x-6) = (x^2+2x+x+2)/(x^2-6x+2x-12) × (x^2-6x-3x+18)/(x^2-3x+2x-6) = (x(x+2) +1(x+2))/(x(x-6) +2(x-6)) × (x(x-6) -3(x-6))/(x(x-3) +2(x-3)) = ((x+2)(x+1))/((x-6)(x+2)) × ((x-6)(x-3))/((x-3)(x+2)) = (x+1)/(x+2)

Example 7: Simplify: (x-4)/(x+4) × (x-3)/(x+4) ÷ (x^2-7x+12)/(x^2+7x+12)
Solution: (x-4)/(x+4) × (x-3)/(x+4) ÷ (x^2-7x+12)/(x^2+7x+12) = (x-4)/(x+4) × (x-3)/(x+4) × (x^2+7x+12)/(x^2-7x+12) = (x-4)/(x+4) × (x-3)/(x+4) × (x^2+3x+4x+12)/(x^2-3x-4x+12) = (x-4)/(x+4) × (x-3)/(x+4) × (x(x+3) +4(x+3))/(x(x-3) -4(x-3)) = (x-4)/(x+4) × (x-3)/(x+4) × ((x+3)(x+4))/((x-3)(x-4)) = (x+3)/(x+4)


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About author: Min Rajbanshi

I am a school math teacher and a blogger since 2012. I do blogging on a variety of topics including math. Over the years, I've explored and written about a wide range of topics, with a particular emphasis on mathematics. My passion for education extends beyond the classroom, as I enjoy sharing knowledge and insights on various subjects through my blog.

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