\(12a^{3}b^{7}(ab)^{-4}=81\)
We can start by simplifying the expression on the left side of the equation. We have
\begin{align*}
12a^{3}b^{7}(ab)^{-4} &= 12a^{3}b^{7}a^{-4}b^{-4} \\
&= 12a^{3+(-4)}b^{7+(-4)} \\
&= 12a^{-1}b^{3}.
\end{align*}
So now our equation is
\[12a^{-1}b^{3} = 81.\]
We can get rid of the fraction by multiplying both sides by \(a\):
\[12b^{3} = 81a.\]
Now we can divide both sides by 12 to isolate the variable \(b^{3}\):
\[b^{3} = \frac{81a}{12} = \frac{27a}{4}.\]
To solve for \(b\), we can take the cube root of both sides:
\[b = \sqrt[3]{\frac{27a}{4}}.\]