$$\frac{\left(3x^{9}\right)^{2}}{\left(4y^{6}\right)^{2}}$$
Expand $\left(3x^{9}\right)^{2}$.$$\frac{3^{2}\left(x^{9}\right)^{2}}{\left(4y^{6}\right)^{2}}$$
To raise a power to another power, multiply the exponents. Multiply $9$ and $2$ to get $18$.$$\frac{3^{2}x^{18}}{\left(4y^{6}\right)^{2}}$$
Calculate $3$ to the power of $2$ and get $9$.$$\frac{9x^{18}}{\left(4y^{6}\right)^{2}}$$
Expand $\left(4y^{6}\right)^{2}$.$$\frac{9x^{18}}{4^{2}\left(y^{6}\right)^{2}}$$
To raise a power to another power, multiply the exponents. Multiply $6$ and $2$ to get $12$.$$\frac{9x^{18}}{4^{2}y^{12}}$$
Calculate $4$ to the power of $2$ and get $16$.$$\frac{9x^{18}}{16y^{12}}$$