The best way to simplify an expression like this is to add all the relevant assumptions to FullSimplify
. For instance, if we use your expression, and assume that both x
and y
are positive, then you get a dramatically simplified result:
expr = ((x^9 - x^6*y^3)^(1/3) - y^2*(8*x^6/y^3 - 8*x^3)^(1/3) +
x*y*(y^3 - y^6/x^3)^(1/3))*(x +
y)/(((x^8)^(1/3)*(x^2 - 2*y^2) + (x^2*y^12)^(1/3))*(1 +
y/x + (y/x)^2)^(1/3));
FullSimplify[expr, {Positive[x], Positive[y]}]
(* (x - y)^(1/3)/(x + y) *)
Oftentimes, if you are experimenting with other functions to help in your quest to find a simpler form for an expression, you can check the language which assumptions they make about the expression and use those with FullSimplify
. For example, the documentation for PowerExpand
says the following:
The transformations made by PowerExpand are correct in general only if $c$ is an integer or $a$ and $b$ are positive real numbers.
Obviously it would be very silly to assume that 1/3 is an integer, but it may be reasonable to assume that x
and y
are positive.
(x^3 - y^3)^(1/3)/((x + y) (x^2 + x y + y^2)^(1/3))
but it’s not until the end $\endgroup$(-1.)^(1/3) (-8.)^(1/3)
is not equal to(8.)^(1/3)
(that is, $a^{1/3}b^{1/3}$ is not equivalent to $(ab)^{1/3}$). $\endgroup$Simplify[expr, 0 < y && 0 < x]
and consider whether the assumption0 < y && 0 < x
is valid in your case. $\endgroup$