Let's convert the decimal number inside the cube root into an algebraic fraction:
$$0.000216=\frac{216}{1000000}=\frac{216}{10^6}$$

Applying the cube root to the numerator and the denominator separately:

$$\sqrt[3]{0.000216}=\frac{\sqrt[3]{216}}{\sqrt[3]{10^6}}$$

We know that $6^3=216$ and $(10^2{)}^3=10^6=1,000,000$.

$$\sqrt[3]{0.000216}=\frac{6}{10^2}=\frac{6}{100}=0.06$$