Recall that dividing something by xx is the same as multiplying it by the inverse 1/x1x. That is, a/b div c/d = a/b * d/cab÷cd=ab⋅dc. 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)x2−16y2x2+3xy−4y2÷x2−8xy+16y2x−y
= ((x^2 - 16y^2)(x-y))/((x^2 + 3xy - 4y^2)(x^2 - 8xy + 16y^2))=(x2−16y2)(x−y)(x2+3xy−4y2)(x2−8xy+16y2)
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)x2−16y2=(x−4y)(x+4y)
x^2 - 8xy + 16y^2 = (x-4y)(x-4y)x2−8xy+16y2=(x−4y)(x−4y)
x^2 + 3xy - 4y^2 = (x+4y)(x - y)x2+3xy−4y2=(x+4y)(x−y)
Making the replacements as needed, this gives
((x-4y)(x+4y)(x-y))/((x+4y)(x-y)(x-4y)^2)(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)=1x−4y⋅x−4yx−4y⋅x+4yx+4y⋅x−yx−y
= 1 / (x - 4y)=1x−4y
Thus, our final answer is 1 / (x-4y)1x−4y.
