[Algebra] Espressioni con prodotti notevoli 3

Semplifica le seguenti espressioni utilizzando, ovunque possibile, i prodotti notevoli.

$\left[(0,\bar{6}x+y)^3-(0,\bar{6}x-y)^3-\left(\frac{3}{2}\right )^{-1}x^2y-2y(x+y)(x-y)+1\right]^2-(4y^3)^2-1=$
$\left[\left(\frac{6}{9}x+y\right )^3-\left(\frac{6}{9}x-y\right )^3-\left(\frac{2}{3}\right )x^2y-2y(x^2-y^2)+1\right]^2-16y^6-1=$
$\left[\left(\frac{\cancel6^2}{\cancel9^3}x+y\right )^3-\left(\frac{\cancel6^2}{\cancel9^3}x-y\right )^3-\left(\frac{2}{3}\right )x^2y-2x^2y+2y^3+1\right]^2-16y^6-1=$
$\left[\left(\frac{2}{3}x+y\right )^3-\left(\frac{2}{3}x-y\right )^3-\left(\frac{2}{3}\right )x^2y-2x^2y+2y^3+1\right]^2-16y^6-1=$
$\left[\frac{8}{27}x^3+y^3+3\left(\frac{4}{9}x^2y\right)+3\left(y^2\frac{2}{3}x\right)-\left(\frac{8}{27}x^3-y^3-3\frac{4}{9}x^2y+3y^2\frac{2}{3}x\right)-\frac{2}{3}x^2y-2x^2y+2y^3+1\right]^2-16y^6-1=$
$\left[\frac{8}{27}x^3+y^3+\not3\frac{4}{\not9^3}x^2y+\not3y^2\frac{2}{\not3}x-\left(\frac{8}{27}x^3-y^3-3\frac{4}{9}x^2y+3y^2\frac{2}{3}x\right)-\frac{2}{3}x^2y-2x^2y+2y^3+1\right]^2-16y^6-1=$
$\left[\frac{8}{27}x^3+y^3+\frac{4}{3}x^2y+2xy^2-\frac{8}{27}x^3+y^3+\frac{4}{3}x^2y-2xy^3-\frac{2}{3}x^2y-2x^2y+2y^3+1\right]^2-16y^6-1=$
$\left[\cancel{\frac{8}{27}x^3-\frac{8}{27}x^3}+\cancel{2xy^2-2xy^2}+y^3+y^3+2y^3+\frac{4}{3}x^2y+\frac{4}{3}x^2y-\frac{2}{3}x^2y-2x^2y+1\right]^2-16y^6-1=$
$\left[y^3+y^3+2y^3+\frac{4}{3}x^2y+\frac{4}{3}x^2y-\frac{2}{3}x^2y-2x^2y+1\right]^2-16y^6-1\\=$
$\left[4y^3+\frac{8}{3}x^2y-\frac{2}{3}x^2y-2x^2y+1\right]^2-16y^6-1=$
$\left[4y^3+\frac{\cancel6^2}{\cancel3^1}x^2y-2x^2y+1\right]^2-16y^6-1=$ $\left[4y^3+2x^2y-2x^2y+1\right]^2-16y^6-1=$
$\left[4y^3+\cancel{2x^2y-2x^2y}+1\right]^2-16y^6-1=$
$\left[4y^3+1\right]^2-16y^6-1=$ $16y^6+1+8y^3-16y^6-1=$
$\cancel{16y^6-16y^6}+\cancel{1-1}+8y^3=$
$8y^3$