1. The problem is to simplify the expression $\sqrt[3]{\frac{40}{27}}$. 2. Recall the property of cube roots: $\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}$. 3. Apply this property to the expression: $$\sqrt[3]{\frac{40}{27}} = \frac{\sqrt[3]{40}}{\sqrt[3]{27}}$$ 4. Simplify the denominator since $27 = 3^3$: $$\frac{\sqrt[3]{40}}{\sqrt[3]{3^3}} = \frac{\sqrt[3]{40}}{3}$$ 5. Factor 40 to simplify the cube root if possible: $$40 = 8 \times 5 = 2^3 \times 5$$ 6. Use the property $\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}$: $$\frac{\sqrt[3]{2^3 \times 5}}{3} = \frac{\sqrt[3]{2^3} \times \sqrt[3]{5}}{3}$$ 7. Simplify $\sqrt[3]{2^3} = 2$: $$\frac{2 \times \sqrt[3]{5}}{3} = \frac{2\sqrt[3]{5}}{3}$$ Final answer: $$\boxed{\frac{2\sqrt[3]{5}}{3}}$$