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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