Find the following product:

To find the product, we will use distributive law as follows:

$0.1 y\left(0.1 x^{5}+0.1 y\right)$

$=(0.1 y)\left(0.1 x^{5}\right)+(0.1 y)(0.1 y)$

$=(0.1 \times 0.1)\left(y \times x^{5}\right)+(0.1 \times 0.1)(y \times y)$

$=(0.1 \times 0.1)\left(x^{5} \times y\right)+(0.1 \times 0.1)\left(y^{1+1}\right)$

$=0.01 x^{5} y+0.01 y^{2}$

Thus, the answer is $0.01 x^{5} y+0.01 y^{2}$.