$$\left(\frac{2y^{-3}x^{6}}{y^{4}}\right)^{-2}$$
To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.$$\left(\frac{2x^{6}}{y^{7}}\right)^{-2}$$
To raise $\frac{2x^{6}}{y^{7}}$ to a power, raise both numerator and denominator to the power and then divide.$$\frac{\left(2x^{6}\right)^{-2}}{\left(y^{7}\right)^{-2}}$$
To raise a power to another power, multiply the exponents. Multiply $7$ and $-2$ to get $-14$.$$\frac{\left(2x^{6}\right)^{-2}}{y^{-14}}$$
Expand $\left(2x^{6}\right)^{-2}$.$$\frac{2^{-2}\left(x^{6}\right)^{-2}}{y^{-14}}$$
To raise a power to another power, multiply the exponents. Multiply $6$ and $-2$ to get $-12$.$$\frac{2^{-2}x^{-12}}{y^{-14}}$$
Calculate $2$ to the power of $-2$ and get $\frac{1}{4}$.$$\frac{\frac{1}{4}x^{-12}}{y^{-14}}$$