Simplifying an expression $\frac{x^7+y^7+z^7}{xyz(x^4+y^4+z^4)}$ if we know $x+y+z=0$ The following expression is given:
$$\frac{x^7+y^7+z^7}{xyz(x^4+y^4+z^4)}$$
Simplify it, knowing that $x+y+z=0$.

Exploit the innate symmetry! Using Newton's identities  to rewrite the power sums as elementary symmetric functions is very simple because $\rm\:e_1 = x+y+z = 0\:$ kills many terms.
Write $\rm\ \ c = e_2 = xy + yz + zx,\ \ \ d = e_3 = xyz,\ \ \ p_k =\: x^k + y^k + z^k.$
$\rm\qquad\qquad p_1\ =\ e_1 = 0$
$\rm\qquad\qquad p_2\ = {-}2\: c$
$\rm\qquad\qquad p_3\ =\ \ \ 3\: d$
$\rm\qquad\qquad p_4\ =  - c\: p_2\ =\ 2\: c^2$
$\rm\qquad\qquad p_5\ = -c\: p_3 +\: d\: p_2\ = {-}5\: c\: d$
$\rm\qquad\qquad p_7\ = -c\: p_5 +\: d\: p_4\ =\ 7\: c^2 d$
Hence $\rm\displaystyle\ \frac{p_7}{p_4 d}\: =\: \frac{7\: c^2\: d}{2\: c^2\: d}\: =\: \frac{7}{2}$
