MULTIPLYING RATIONAL EXPRESSIONS WORKSHEET

Multiply the following rational expressions and show the product in simplest form :

1.  [(x² - 2x)/(2x + 4)] ⋅ [(3x + 6)/(5x - 10)]

2.  [(6x + 6)/(x² - 4)] ⋅ [(x² + 6x + 8)/(3x + 3)]

3.  [(a + b)/(a - b)] ⋅ [(a³ - b³)/(a³ + b³)]

4.  (5ab/15cd) ⋅ (4bc/32ad) ⋅ (16ac/2bc)

5.  [(x² - 2x + 1)/(x² - 3x + 2)] ⋅ [(3x - 6)/(6x - 6)]

6.  [(x² - 25)/(x + 3)] ⋅ [(x² - 9)/(x + 5)²]

7.  [(x² - 9y²)/(3x - 3y)] ⋅ [(x - y)/(x + 3y)]

8.  [(x² - 16)/(x - 2)] ⋅ [(x - 2)/(x³ + 64)]

9.  [(p² - 1)/p] ⋅ [p³/(p - 1)] ⋅ [1/(p + 1)]

10.  [(px - 2p)/(qx - 3q)] ⋅ [(bx - 3b)/(ax - 2a)]

11. [(3b² + 18b)/(b + 6)] ⋅ [1/(b + 8)]

12.  [(p + 7)/(p - 10)] ⋅ [(p + 2)/(7p + 14)]

13.  [1/(p - 9)] ⋅ [(p² + 6p - 27)/(p + 9)]

14.  [(x² - 10x + 25) / (10x - 100)] ⋅ [(x - 10)/(45 - 9x)]

15.  [45x² / (x - 9)] ⋅[(x² - 5x - 36) / (3x³ + 12x²)]

1. Answer :

=  [(x² - 2x)/(2x + 4)] ⋅ [(3x + 6)/(5x - 10)]

Multiply numerators and denominators. 

=  [(x² - 2x)(3x + 6)] /[(2x + 4)(5x - 10)]

Factor the numerator and denominator. 

=  [x(x - 2)3(x + 2)] /[2(x + 2)5(x - 2)]

=  [3x(x - 2)(x + 2)] /[10(x + 2)(x - 2)]

Cancel common factors.  

=  3x/10

2. Answer :

=  [(6x + 6)/(x² - 4)] ⋅ [(x² + 6x + 8)/(3x + 3)]

Multiply numerators and denominators. 

=  [(6x + 6)(x² + 6x + 8)] / [(x² - 4)(3x + 3)]

Factor the numerator and denominator. 

=  [6(x + 1)(x + 2)(x + 4)] / [(x + 4)(x - 4)3(x + 1)]

=  [6(x + 1)(x + 2)(x + 4)] / [3(x + 4)(x - 4)(x + 1)]

Cancel common factors. 

=  [6(x + 2)] / [3(x - 4)]

=  2(x + 2)/(x - 4)

3. Answer :

=  [(a + b)/(a - b)] ⋅ [(a³ - b³)/(a³ + b³)]

Multiply numerators and denominators. 

=  [(a + b)(a³ - b³)]/[(a - b)(a³ + b³)]

Factor the numerator and denominator. 

=  [(a + b)(a - b)(a² + ab + b²)]/[(a - b)(a + b)(a² - ab + b²)]

Cancel common factors. 

=  (a² + ab + b²) / (a² - ab + b²)

4. Answer :

=  (5ab/15cd) ⋅ (4bc/32ad) ⋅ (16ac/2bc)

Multiply numerators and denominators. 

=  (5ab ⋅ 4bc ⋅ 16ac) / (15cd ⋅ 32ad ⋅ 2bc)

=  320a²b²c² / 960abc²d²

Cancel common factors. 

=  ab/3d²

5. Answer :

=  [(x² - 2x + 1)/(x² - 3x + 2)] ⋅ [(3x - 6)/(6x - 6)]

Multiply numerators and denominators. 

[(x² - 2x + 1)(3x - 6)] / [(x² - 3x + 2)(6x - 6)]

Factor the numerator and denominator. 

=  [(x - 1)(x - 1)3(x - 2)] / [(x - 2)(x - 1)6(x - 1)]

=  [3(x - 1)(x - 1)(x - 2)] / [6(x - 2)(x - 1)(x - 1)]

Cancel common factors. 

=  3/6

=  1/2

6. Answer :

=  [(x² - 25)/(x + 3)] ⋅ [(x² - 9)/(x + 5)²]

Multiply numerators and denominators. 

=  [(x² - 25)(x² - 9)] ⋅ [(x + 3)(x + 5)²]

Factor the numerator and denominator. 

=  [(x + 5)(x - 5)(x + 3)(x - 3)] / [(x + 3)(x + 5)(x + 5)]

Cancel common factors. 

=  [(x - 5)(x - 3)] / (x + 5)

7. Answer :

=  [(x² - 9y²)/(3x - 3y)] ⋅ [(x - y)/(x + 3y)]

Multiply numerators and denominators. 

=  [(x² - 9y²)(x - y)] / [(3x - 3y)(x + 3y)]

Factor the numerator and denominator. 

=  [(x + 3y)(x - 3y)(x - y)] / [3(x - y)(x + 3y)]

Cancel common factors. 

=  (x - 3y)/3

8. Answer :

=  [(x² - 16)/(x - 2)] ⋅ [(x - 2)/(x³ + 64)]

Multiply numerators and denominators. 

=  [(x² - 16)(x - 2)] / [(x - 2)(x³ + 64)]

=  [(x² - 4²)(x - 2)] / [(x - 2)(x³ + 4³)]

Factor the numerator and denominator. 

=  [(x + 4)(x - 4)(x - 2)] / [(x - 2)(x + 4)(x² - 4x + 16)]

Cancel common factors. 

=  (x - 4)/(x² - 4x + 16)

9. Answer :

=  [(p² - 1)/p] ⋅ [p³/(p - 1)] ⋅ [1/(p + 1)]

Multiply numerators and denominators. 

=  [(p² - 1) ⋅ p³ ⋅ 1] / [p(p - 1)(p + 1)]

=  p³(p² - 1) / p(p² - 1²)

=  p³(p² - 1) / p(p² - 1)

Cancel common factors. 

=  p²

10. Answer :

=  [(px - 2p)/(qx - 3q)] ⋅ [(bx - 3b)/(ax - 2a)]

Multiply numerators and denominators. 

=  [(px - 2p)(bx - 3b)] / [(qx - 3q)(ax - 2a)]

Factor the numerator and denominator. 

=  [p(x - 2)b(x - 3)] / [q(x - 3)a(x - 2)]

=  [bp(x - 2)(x - 3)] / [aq(x - 3)(x - 2)]

Cancel common factors. 

=  bp/aq

11. Answer :

= [(3b² + 18b)/(b + 6)] ⋅ [1/(b + 8)] ----(1)

3b² + 18b = 3b(b + 6)

Applying the factored form in (1), we get

= [3b(b + 6)/(b + 6)] ⋅ [1/(b + 8)]

= 3b/(b + 8)

12. Answer :

= [(p + 7)/(p - 10)] ⋅ [(p + 2)/(7p + 14)]

= [(p + 7)/(p - 10)] ⋅ [(p + 2)/7(p + 2)]

= (p + 7) / 7(p - 10)

13. Answer :

= [1/(p - 9)] ⋅ [(p² + 6p - 27)/(p + 9)]

p² + 6p - 27 = p² + 9p - 3p - 27

= p(p + 9) - 3(p + 9)

= (p + 9)(p - 3)

= [1/(p - 9)] ⋅ [(p + 9)(p - 3)/(p + 9)]

= 1/(p - 9)(p - 3)

14. Answer :

= [(x² - 10x + 25) / (10x - 100)] ⋅ [(x - 10)/(45 - 9x)]

x² - 10x + 25 = x² - 5x - 5x + 25

= x(x - 5) - 5(x - 5)

= (x - 5)(x - 5)

10x - 100 = 10(x - 10)

45 - 9x = 9(5 - x)

Factoring -9, we get

= -9(x - 5)

= [ (x - 5)(x - 5)/ 10(x - 10)] ⋅ [-(x - 10)/(x - 5)]

= -(x - 5) / 10

15. Answer :

= [45x² / (x - 9)] ⋅[(x² - 5x - 36) / (3x³ + 12x²)]

= [45x² / (x - 9)] ⋅[(x - 9)(x + 4) / 3x²(x + 4)]

= 15

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