# Simplify [2/xy + x/(x^2*y)] / [ (x-y)/xy^2 + (x-xy)/x^2*y^2]

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To simplify [2/xy + x/(x^2*y)] / [ (x-y)/xy^2 + (x-xy)/x^2*y^2]

, we multiply both numerator and denominator by x^2y^2 .

Numerator *x^2y^2 = 2x^2y^2/xy +x*x^2y^2/x^2y

Numerator *x^2y^2 = 2xy+xy

Numerator*x^2y^2 = 3xy...................(1)

Denominator*x^2y^2 = (x-y)x^2y^2/xy^2 +(x-xy)x^2y^2/x^2y^2.

Denominator*x^2y^2 = (x-y)x +(x-xy) = x^2-xy +x-xy.

Denominator*x^2y^2= x^2+x-2xy .

Denominator*x^2y^2= x(x-2y+1)......(2).

We use the simplified results at (1) and (2) to rewite the given rational fraction:

[2/xy + x/(x^2*y)] / [ (x-y)/xy^2 + (x-xy)/x^2*y^2] = 3xy/x(x-2y+1.)

[2/xy + x/(x^2*y)] / [ (x-y)/xy^2 + (x-xy)/x^2*y^2] = 3y/(x-2y+1).

To simplify the given expression:

[2/xy + x/(x^2*y)] / [ (x-y)/xy^2 + (x-xy)/x^2*y^2]

make the denominator of all the terms the same. Here we see that will be x^2*y^2

=> [2xy/x^2*y^2 + xy/(x^2*y^2)] / [ x(x-y)/x^2y^2 + (x-xy)/x^2*y^2]

cancel x^2*y^2

=> [2xy + xy] / [ x(x-y) + (x-xy)]

=> [2xy + xy] / [ x^2-xy + x - xy]

=> [3xy] / [ x^2 + x -2xy]

cancel x

=> 3y / (x – 2y +1)

**Therefore the result is 3y / (x – 2y +1)**