$\frac{x^{3n} - x^ny^{2n}}{3x^{3n} + 6x^{2n}y^n + 3x^ny^{2n}} = \frac{x^n(x^{2n} - y^{2n})}{3x^n(x^{2n} + 2x^ny^n + y^{2n})} = \frac{(x^n - y^n)(x^n + y^n)}{3(x^n + y^n)^2} = \frac{x^n - y^n}{3(x^n + y^n)}$

$\frac{a^{3n - 1}b^{n + 1} - 4a^{n - 1}b^{n + 1}}{4a^nb^{n - 1} - 4a^{2n}b^{n - 1} + a^{3n}b^{n - 1}} = \frac{a^{n - 1}b^{n + 1}(a^{2n} - 4)}{a^nb^{n - 1}(4 - 4a^n + a^{2n})} = \frac{b^2(a^{n} - 2)(a^{n} + 2)}{a(2 - a^n)^2} = \frac{b^2(a^{n} - 2)(a^{n} + 2)}{a(a^n - 2)^2} = \frac{b^2(a^{n} + 2)}{a(a^n - 2)}$

$\frac{2a^{n + 1} - 4a^{2n + 1} + 2a^{3n + 1}}{4a^{3n} - 4a^n} = \frac{2a^{n + 1}(1- 2a^n + a^{2n})}{4a^n(a^{2n} - 1)} = \frac{a(1 - a^n)^2}{2(a^n - 1)(a^n + 1)} = \frac{a(a^n - 1)^2}{2(a^n - 1)(a^n + 1)} = \frac{a(a^n - 1)}{2(a^n + 1)}$

$\frac{54xy^{3n}z^n - 72x^{n + 1}y^{2n}z^n + 24x^{2n + 1}y^nz^n}{12x^{2n + 2}y^{n - 1}z^{n + 1} - 27x^2y^{3n - 1}z^{n + 1}} = \frac{6xy^nz^n(9y^{2n} - 12xy + 4x^{2n})}{3x^2y^{n - 1}z^{n + 1}(4x^{2n} - 9y^{2n})} = \frac{2y(3y^{n} - 2x^{n})^2}{xz(2x^{n} - 3y^{n})(2x^{n}+ 3y^{n})} = \frac{2y(2x^{n} - 3y^{n})^2}{xz(2x^{n} - 3y^{n})(2x^{n}+ 3y^{n})} = \frac{2y(2x^{n} - 3y^{n})}{xz(2x^{n}+ 3y^{n})}$