Question

How do you simplify `((x/y) - (4/x)) / ((1/x) - (2/y^2))`?

Answer 1

`(y(x^2-4y))/(y^2-2x)`

Explanation:

Before you can do the division of the numerator by the denominator, you have to have one fraction at the top and one at the bottom.
The calculation can be shown as:

`"numerator " div " denominator"`

`" "(x/y - 4/x) div (1/x - 2/y^2)" "larr` find common denominators

`(x/y xx color(red)(x/x) - 4/x xx color(red)(y/y)) div (1/x xx color(red)(y^2/y^2) - 2/y^2xx color(red)(x/x))`

Each fraction has been `color(red)("multiplied by" 1)`

`(x^2-4y)/(xy) div (y^2-2x)/(xy^2)`

To divide fractions, multiply by the reciprocal of the second fraction:

`(x^2-4y)/(xy) div (xy^2)/(y^2-2x)" "larr` now simplify:

`(y(x^2-4y))/(y^2-2x)`