SIMPLIFYING RATIONAL EXPRESSIONS EXAMPLES

Example 1 :

(i)  Simplify

(4x² y / 2z²) ⋅ (6xz³/20y⁴)

Solution :

  =  (4x² y / 2z²) ⋅ (6xz³/20y⁴)

  =  (x/2) ⋅ (3xz/5y³)

  =  3x²z/10y³

(ii)  (p² - 10p + 21)/(p - 7)⋅ (p² + p - 12)/(p-3)²

Solution :

  =  (p² - 10p + 21)/(p - 7) ⋅ (p² + p - 12)/(p-3)²

p² - 10p + 21  =  (p - 7)(p - 3)

p² + p - 12  =  (p + 4)(p - 3)

  =  [(p - 7)(p - 3)/(p - 7)] ⋅ [(p + 4)(p - 3)/(p-3)²]

=  (p - 3) (p - 3) (p + 4)/(p-3)²

=  p + 4

Hence the answer is p + 4.

(iii)  [5t³/(4t - 8)] ⋅ [(6t - 12)/10t]

Solution :

  =   [5t³/(4t - 8)] ⋅ [(6t - 12)/10t]

4t - 8  =  4(t - 2)

6t - 12  =  6(t - 2)

  =   [5t³/4(t - 2)] ⋅ [6(t - 2)/10t]

  =  6t²/8

  =  3t²/4

Example 2 :

Simplify

(i) [(x + 4)/(3x + 4y)] ⋅ [(9x² - 16y²)/(2x² + 3x - 20)]

Solution :

  =  [(x + 4)/(3x + 4y)] ⋅ [(9x² - 16y²)/(2x² + 3x - 20)]

(9x² - 16y²)  =  (3x)² - (4y)²

  =  (3x + 4y) (3x - 4y)

2x² + 3x - 20  =  (x + 4)(2x - 5)

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

  =  (3x - 4y)/(2x - 5)

(ii) [(x³ - y³)/(3x²+9xy+6y²) ⋅ (x² + 2xy + y²)/(x² - y²)

Solution :

  =  [(x³ - y³)/(3x²+9xy+6y²)] ⋅ [(x² + 2xy + y²)/(x² - y²)]

x³ - y³  =  (x - y)(x² + xy + y²)

3x²+9xy+6y²   =  3x² + 6xy + 3xy + 6y²

  =  3x(x + 2y) + 3y(x + 2y)

  =  3(x + y) (x + 2y)

  =  [(x-y)(x²+xy+y²)/3(x+y) (x+2y)] ⋅ [(x+y)²/(x+y)(x-y)]

=  (x²+xy+y²)/3(x+2y)

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