To generate an equivalent algebraic expression to $\left(\frac{4x^4}{y^4}\right)^3 \div \left(\frac{x^7}{y^3}\right)^2$, we can apply the properties of integer exponents.
First, let's simplify the expression inside the parentheses:
$\left(\frac{4x^4}{y^4}\right)^3 = \frac{(4x^4)^3}{(y^4)^3}$
$\left(\frac{x^7}{y^3}\right)^2 = \frac{(x^7)^2}{(y^3)^2}$
Now, we can apply the rules of exponents:
$\frac{(4x^4)^3}{(y^4)^3} = \frac{4^3 x^{4 \times 3}}{y^{4 \times 3}} = \frac{64x^{12}}{y^{12}}$
$\frac{(x^7)^2}{(y^3)^2} = \frac{x^{7 \times 2}}{y^{3 \times 2}} = \frac{x^{14}}{y^{6}}$
Next, we divide the two expressions:
\[\frac{\frac{64x^{12}}{y^{12}}}{\frac{x^{14}}{y^{6}}} = \frac{64x^{12}}{y^{12}} \cdot \frac{y^{6}}{x^{14}}\]
Finally, we simplify the expression further:
\[\frac{64x^{12}}{y^{12}} \cdot \frac{y^{6}}{x^{14}} = 64x^{12} \cdot y^{6} \cdot \frac{1}{y^{12}} \cdot \frac{1}{x^{14}} = \frac{64x^{12} \cdot y^{6}}{x^{14} \cdot y^{12}} = \frac{64y^{6}}{x^{2}}.\]
Therefore, an equivalent algebraic expression to $\left(\frac{4x^4}{y^4}\right)^3 \div \left(\frac{x^7}{y^3}\right)^2$ is $\frac{64y^{6}}{x^{2}}$.