Question

How do you simplify `\frac { x ^ { 2} - 16y ^ { 2} } { x ^ { 2} + 3x y - 4y ^ { 2} } \div \frac { x ^ { 2} - 8x y + 16y ^ { 2} } { x - y }`?

Answer 1

`1/ (x - 4y)`

Explanation:

Recall that dividing something by `x` is the same as multiplying it by the inverse `1/x`. That is, `a/b div c/d = a/b * d/c`. We use this algebraic fact to help us simplify.

`(x^2 - 16y^2)/(x^2 + 3xy - 4y^2) div (x^2 - 8xy + 16y^2)/(x-y)`
`= ((x^2 - 16y^2)(x-y))/((x^2 + 3xy - 4y^2)(x^2 - 8xy + 16y^2))`

Now we note that most of these terms can be factored. See that the following are true:

`x^2 - 16y^2 = (x-4y)(x+4y)`
`x^2 - 8xy + 16y^2 = (x-4y)(x-4y)`
`x^2 + 3xy - 4y^2 = (x+4y)(x - y)`

Making the replacements as needed, this gives

`((x-4y)(x+4y)(x-y))/((x+4y)(x-y)(x-4y)^2)`
`= 1/(x-4y) * (x-4y)/(x-4y) * (x+4y)/(x+4y) * (x-y)/(x-y)`
`= 1 / (x - 4y)`

Thus, our final answer is `1 / (x-4y)`.