$$\frac{\left(2yx^{-3}\right)^{3}}{x^{3}y^{2}\times 2y^{3}}$$
To multiply powers of the same base, add their exponents. Add $2$ and $3$ to get $5$.$$\frac{\left(2yx^{-3}\right)^{3}}{x^{3}y^{5}\times 2}$$
Expand $\left(2yx^{-3}\right)^{3}$.$$\frac{2^{3}y^{3}\left(x^{-3}\right)^{3}}{x^{3}y^{5}\times 2}$$
To raise a power to another power, multiply the exponents. Multiply $-3$ and $3$ to get $-9$.$$\frac{2^{3}y^{3}x^{-9}}{x^{3}y^{5}\times 2}$$
Calculate $2$ to the power of $3$ and get $8$.$$\frac{8y^{3}x^{-9}}{x^{3}y^{5}\times 2}$$
Cancel out $2y^{3}$ in both numerator and denominator.$$\frac{4x^{-9}}{y^{2}x^{3}}$$
To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.$$\frac{4}{y^{2}x^{12}}$$