SIMPLIYING RATIONAL EXPRESSION WITH MULTUIPLICATION AND DIVISION

Problem 1 :

(x² + 4x – 12)/[x²(x² + 9x + 18)] ⋅ (6x)

Solution :

(x² + 4x – 12)/[x²(x² + 9x + 18)] ⋅ (6x)

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

= (x – 2)[x²(x + 3)] ⋅ (6x)

= (x – 2)/[x(x + 3)] ⋅ (6)

= 6(x – 2)/(x(x + 3))

Problem 2 :

(3x² – 12)/(5x – 10) ⋅ [1/(2x + 4)]

Solution :

(3x² – 12)/(5x – 10) ⋅ [1/(2x + 4)]

= 3(x² – 4)/5(x – 2) ⋅ [1/(2x + 4)]

= 3(x² – 2²)/5(x – 2) ⋅ [1/(2(x + 2))]

= (3(x – 2) (x + 2))/5(x – 2) ⋅ [1/(2(x + 2))]

= 3/10

Problem 3 :

(x² – 4)/(x² + 4) ⋅ [(x + 2)/(x – 2)]

Solution :

(x² – 4)/(x² + 4) ⋅ [(x + 2)/(x – 2)]

= (x² – 2²)/(x² + 4) ⋅ [(x + 2)/(x – 2)]

= ((x + 2) (x – 2))/(x² + 4) ⋅ [(x + 2)/(x – 2)]

= (x + 2)²/(x² + 4)

Problem 4 :

(5x² – 20)/25x² ⋅ [x/(x - 2)]

Solution :

(5x² – 20)/25x² ⋅ [x/(x - 2)]

 = 5(x² – 4)/25x² ⋅ [x/(x - 2)]

= 5(x² – 2²)/25x² ⋅ [x/(x - 2)]

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

= (x + 2)/5x

Problem 5 :

12x²y³z/6x³y²2²

Solution :

12x²y³z/6x³y²2²

= 12x²y³z/6x³y² × 4

= 12x²y³z/24x³y²

= yz/2x

Problem 6 :

[(x³ + 3x²)/2x] ÷ [(x² + 5x + 6)]/5x³

Solution :

[(x³ + 3x²)/2x] ÷ [(x² + 5x + 6)]/5x³

= [x²(x + 3)/2x] ÷ [(x² + 5x + 6)]/5x³

= [x(x + 3)/2] ÷ [(x² + 2x + 3x + 6)]/5x³

= [x(x + 3)/2] ÷ [(x² + 2x) + (3x + 6)]/5x³

= [x(x + 3)/2] ÷ [x(x + 2) + 3(x + 2)]/5x³

= [x(x + 3)/2] ÷ [(x + 3) (x + 2)]/5x³ 

= [x(x + 3)/2] × [5x³/((x + 3) (x + 2))]

= 5x⁴/2(x + 2)

Problem 7 :

[(x² + x – 20)/(x + 1)] ÷ [(11x + 55)/(x + 1)]

Solution :

[(x² + x – 20)/(x + 1)] ÷ [(11x + 55)/(x + 1)]

= [(x² – 4x + 5x – 20)/(x + 1)] ÷ [(11x + 55)/(x + 1)]

= [(x(x – 4) + 5(x - 4))/(x + 1)] × [(x + 1)/(11x + 55)]

= [((x + 5) (x – 4))/(x + 1)] × [(x + 1)/11(x + 5)]

= (x – 4)/11

Problem 8 :

[(x² + 5x + 6)/(x + 3)] ÷ [(x² – 4)/(x + 1)]

Solution :

[(x² + 5x + 6)/(x + 3)] ÷ [(x² – 4)/(x + 1)]

= [(x² + 3x + 2x + 6)/(x + 3)] ÷ [(x² – 4)/(x + 1)]

= [(x(x + 3) + 2(x + 3))/(x + 3)] ÷ [(x² – 2²)/(x + 1)]

= [((x + 3) (x + 2))/(x + 3)] ÷ [((x + 2) (x – 2))/(x + 1)]

= [((x + 3) (x + 2))/(x + 3)] × [(x + 1)/((x + 2) (x – 2)]

= (x + 1)/(x – 2)