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

Question

Kennedy Adams · Accepted Answer

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

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)#

Savannah Anderson · Answer

undefined To simplify the expression ((x/y) - (4/x)) / ((1/x) - (2/y^2)), we can follow these steps:

1. Find a common denominator for each fraction within the expression.
2. Simplify each fraction by multiplying the numerator and denominator by the common denominator.
3. Combine the fractions by subtracting the numerators over the common denominator.
4. Simplify the resulting fraction if possible.

Let's go through the steps:

1. The common denominator for the first fraction (x/y) and the second fraction (4/x) is xy.
2. Simplifying the first fraction: ((x/y) * (xy)) / (xy) = (x^2) / xy.
   Simplifying the second fraction: ((4/x) * (xy)) / (xy) = (4y) / xy.
3. Combining the fractions: ((x^2) / xy) - ((4y) / xy) = (x^2 - 4y) / xy.
4. The resulting simplified expression is (x^2 - 4y) / xy.