Factorise

$x^{9}-y^{9}=\left(x^{3}\right)^{3}-\left(y^{3}\right)^{3}$

we know

$a^{3}-b^{3}=(a-b)\left(a^{2}+b^{2}+a b\right)$

$a=x^{3}, b=y^{3}$

So, $x^{9}-y^{9}=\left(x^{3}\right)^{3}-\left(y^{3}\right)^{3}=\left(x^{3}-y^{3}\right)\left(x^{6}+y^{6}+x^{3} y^{3}\right)$

$=(x-y)\left(x^{2}+y^{2}+x y\right)\left(x^{6}+y^{6}+x^{3} y^{3}\right)$