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   =  3x+ 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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