<s>[INST] 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 $x+y+z=0$.
 [/INST] Use that $z=-(x+y)$, so you have that the numerator turns out to be $$x^7+y^7-(x+y)^7=-7x^6y-21x^5y^2-35x^4y^3-35x^3y^4-21x^2y^5-7xy^6$$Also, the denominator turns out to be $$xy(-x-y)(x^4+y^4+(-x-y))^4=-2x^6y-6x^5y^2-10x^4y^3-10x^3y^4-6x^2t^5-2xy^6$$You can factor a $7$ from the numerator and a $2$ from the denominator, and the answer turns out to be $$\frac{7}{2}$$
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