Let's simplify the given expression step by step:

\((\sqrt[3]{64a^{3}})^{-1}\)

First, let's simplify the cube root of \(64a^3\):

\(\sqrt[3]{64a^{3}} = 4a\)

Now we have:

\((4a)^{-1}\)

Recall that \(x^{-1} = \frac{1}{x}\):

\(\frac{1}{4a}\)

Therefore, the simplified form of \((\sqrt[3]{64a^{3}})^{-1}\) is \(\frac{1}{4a}\).