Multiply:

To multiply, we will use distributive law as follows:

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

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

$=\left(x^{8}+x^{6} y^{2}\right)-\left(y^{6} x^{2}+y^{8}\right)$

$=x^{8}+x^{6} y^{2}-y^{6} x^{2}-y^{8}$

Thus, the answer is $x^{8}+x^{6} y^{2}-y^{6} x^{2}-y^{8}$.