MULTIPLYING AND DIVIDING RATIONAL EXPRESSIONS WORKSHEET

Problems 1-5 : Perform the indicated multiplication and simplify :

Problem 1 : 

(x³/9y²) ⋅ (27y/x⁵)

Problem 2 : 

[(x² + 2x - 3)/(x² + 8x + 16)] ⋅ [(3x + 12)/(x - 1)]

Problem 3 : 

[4x/(x² - 4)] ⋅ [(x + 2)/16x]

Problem 4 : 

[(x² - 25)/(x² - 16)] ⋅ [(x + 4)/(x + 5)]

Problem 5 : 

[(x² - x - 12)/(x² - 9)] ⋅ [(3 + x)/(4 - x)]

Problems 1-5 : Perform the indicated division and simplify :

Problem 6 : 

(14x⁴/y) ÷ (7x/3y⁴)

Problem 7 : 

[6/(28x + 4)] ÷ [6/(35x + 5)]

Problem 8 : 

[4x/(x - 6)] ÷ [4x/(8x - 48)]

Problem 9 : 

[7x²/(7x³ + 56x²)] ÷ [2 / (x² + 7x - 8)]

Problem 10 : 

[(x - 4)/(x² - 4)] ÷ [(x² - 3x - 4)/(x² + 5x + 6)]

Problem 11 : 

[(x² - 16)/(x² - 10x + 25)] ÷ [(3x - 12)/(x² - 3x - 10)]

Problem 12 :

[(x² - y²)/(4x + 4y)] ÷ [(3y - 3x)/12x²]

Problem 13 :

[(y² - 36)/(y² - 8y + 16)] ÷ [(3y - 18)/(y² - y - 12)]

Detailed Answer Key

1. Answer : 

=  (x³/9y²) ⋅ (27y / x⁵)

=  (x³ ⋅ 27y) / (9y² ⋅ x⁵)

=  3/ x²y

2. Answer : 

=  [(x² + 2x - 3)/(x² + 8x + 16)] ⋅ [(3x + 12)/(x - 1)]

Factor.

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

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

Cancel common factors. 

=  3(x + 3) / (x + 4)

3. Answer : 

=  [4x/(x² - 4)] ⋅ [(x + 2)/16x]

=  [4x(x + 2)] / [16x(x² - 4)]

Factor.

=  [4x(x + 2)] / [16x(x + 2)(x - 2)]

Cancel common factors. 

=  1/[4(x + 2)]

4. Answer : 

=  [(x² - 25)/(x² - 16)] ⋅ [(x + 4)/(x + 5)]

=  [(x² - 25)(x + 4)]//[(x² - 16)(x + 5)]

Factor.

=  [(x + 5)(x - 5)(x + 4)] ⋅ [(x + 4)(x - 4)(x + 5)]

Cancel common factors. 

=  (x - 5)/(x - 4)

5. Answer : 

=  [(x² - x - 12)/(x² - 9)] ⋅ [(3 + x)/(4 - x)]

=  [(x² - x - 12)(3 + x)] ⋅ [(x² - 9)(4 - x)]

Factor. 

=  [(x - 4)(x + 3)(x + 3)] ⋅ [(x + 3)(x - 3)(4 - x)]

=  [(x - 4)(x + 3)(x + 3)] ⋅ [-(x + 3)(x - 3)(x - 4)]

Cancel common factors. 

=  (x + 3)/[-(x - 3)]

=  (x + 3)/(3 - x)

6. Answer : 

=  (14x⁴/y) ÷ (7x/3y⁴)

Invert and multiply.

=  (14x⁴/ y) ⋅ (3y⁴/7x)

=  (14x⁴ ⋅ 3y⁴) / (7x ⋅ y)

Cancel common factors. 

=  6x³y³

7. Answer :

=  [6/(28x + 4)] ÷ [6/(35x + 5)]

Invert and multiply.

=  [6/(28x + 4)] ⋅ [(35x + 5)/6]

=  [6(35x + 5)] / [6(28x + 4)]

Simplify. 

=  (35x + 5)/(28x + 4)

Factor. 

=  [5(7x + 1)]/[4(7x + 1)]

Cancel common factors. 

=  5/4

8. Answer :

=  [4x/(x - 6)] ÷ [4x/(8x - 48)]

Invert and multiply.

=  =  [4x/(x - 6)] ⋅ [(8x - 48)/4x]

=  [4x(8x - 48)] / [4x(x - 6)]

Simplify. 

=  (8x - 48)/(x - 6)

Factor. 

=  8(x - 6)/(x - 6)

Cancel common factors. 

=  8

9. Answer : 

=  [7x²/(7x³ + 56x²)] ÷ [2 / (x² + 7x - 8)]

Invert and multiply.

=  [7x²/(7x³ + 56x²)] ⋅ [(x² + 7x - 8)/2]

=  [7x²(x² + 7x - 8)] / [2(7x³ + 56x²)]

Factor. 

=  [7x²(x - 1)(x + 8)] / [14x²(x + 8)]

Cancel common factors. 

=  (x - 1)/2

10. Answer : 

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

Invert and multiply.

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

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

Factor.

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

Cancel common factors. 

=  (x + 3)/[(x - 2)(x + 1)]

11. Answer : 

= [(x² - 16)/(x² - 10x + 25)] ÷ [(3x - 12)/(x² - 3x - 10)]

x² - 16 = x² - 4²

= (x - 4)(x + 4)

x² - 10x + 25 = x² - 5x - 5x + 25

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

= (x - 5)(x - 5)

3x - 12 = 3(x - 4)

x² - 3x - 10 = x² - 5x + 2x - 10

= x(x - 5) + 2(x - 5)

= (x + 2)(x - 5)

= [(x - 4)(x + 4)/(x - 5)(x - 5)] ÷ [3(x - 4)/(x + 2)(x - 5)]

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

= (x + 4)(x + 2)/3(x - 5)

12. Answer :

= [(x² - y²)/(4x + 4y)] ÷ [(3y - 3x)/12x²]

x² - y² = (x + y)(x - y)

4x + 4y = 4(x + y)

3y - 3x = 3(y - x)

= [(x + y)(x - y)/4(x + y)] ÷ [3(y - x)/12x²]

= [(x + y)(x - y)/4(x + y)] x [12x²/3(y - x)]

= [(x + y)(x - y)/4(x + y)] x [-12x²/3(x - y)]

= -x²

13. Answer :

= [(y² - 36)/(y² - 8y + 16)] ÷ [(3y - 18)/(y² - y - 12)]

y² - 36 = y² - 6²

= (y - 6)(y + 6)

y² - 8y + 16 = y² - 4y - 4y + 16

= y(y - 4) - 4(y - 4)

= (y - 4)(y - 4)

3y - 18 = 3(y - 6)

y² - y - 12 = y² - 4y + 3y - 12

= y (y - 4) + 3(y - 4)

= (y - 4)(y + 3)

= [(y - 6)(y + 6) / (y - 4)(y - 4)] ÷ [3(y - 6)/(y - 4)(y + 3)]

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

= (y + 6)(y + 3)/3(y - 4)

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