Simplify Radical Fraction Expression: (y^-3 - x^-3) / (xy^-1 + x^-1y + 1)

[Agent_J]

Simplify (y^-3 - x^-3) / (xy^-1 + x^-1y + 1)

better picture of it here
http://members.rogers.com/agentj/images/math2.jpg

I tried flipping the variables with negative exponents to the numerator and denominator, but then had no idea what to do next

 

[AntiMagicMan]

I doubt you will get better simplification than that, you could multiply top and bottom by xy to get a nice denominator.

 

[cookiemonster]

It certainly does get simpler!

First consider the numerator:

y^-3 - x^-3 = 1/y^3 - 1/x^3 = (x^3 - y^3)/(x^3 y^3) = (x - y)(x^2 + y^2 + xy)/(x^3 y^3)

Now the denominator:

x y^-1 + y x^-1 + 1 = x/y + y/x + 1 = (x^2 + y^2 + xy)/(xy)

Combine them:

(x - y)(x^2 + y^2 + xy)/(x^3 y^3)(xy)/(x^2 + y^2 + xy)

The x^2 + y^2 + xy terms on top and bottom cancel, and one of each of the powers of x and y on the bottom cancel to yield a final simplification of:

$$\frac{x - y}{x^2y^2}$$

Much nicer!

cookiemonster