1. Problem: Simplify the expression $\sqrt[3]{4^{1-5}}$. 2. Compute the exponent inside the power: $1-5=-4$. 3. Substitute the result: $\sqrt[3]{4^{1-5}}=\sqrt[3]{4^{-4}}$. 4. Use the power rule to convert the root to a power: $\sqrt[3]{4^{-4}}=(4^{-4})^{1/3}=4^{-4/3}$. 5. Rewrite the base $4$ as $2^2$ and simplify the exponent: $4^{-4/3}=(2^2)^{-4/3}=2^{-8/3}$. 6. Express with a positive exponent in the denominator and as a radical: $2^{-8/3}=\dfrac{1}{2^{8/3}}=\dfrac{1}{\sqrt[3]{2^8}}=\dfrac{1}{\sqrt[3]{256}}$. 7. Final answer: $\sqrt[3]{4^{1-5}}=2^{-8/3}=\dfrac{1}{\sqrt[3]{256}}$.