1. The problem is to simplify the expression $$\sqrt[3]{32}$$ and evaluate $$\left(\frac{125}{1000}\right)^{\frac{1}{3}}$$. 2. Recall that the cube root of a number $a$ is the number $b$ such that $b^3 = a$. 3. First, simplify $$\sqrt[3]{32}$$. Since $32 = 2^5$, we have: $$\sqrt[3]{32} = \sqrt[3]{2^5} = 2^{\frac{5}{3}} = 2^{1 + \frac{2}{3}} = 2 \times 2^{\frac{2}{3}}$$ 4. Next, simplify $$\left(\frac{125}{1000}\right)^{\frac{1}{3}}$$. 5. Simplify the fraction inside the parentheses: $$\frac{125}{1000} = \frac{\cancel{125}^1}{\cancel{125}^8} = \frac{1}{8}$$ 6. Now evaluate the cube root: $$\left(\frac{1}{8}\right)^{\frac{1}{3}} = \frac{1^{\frac{1}{3}}}{8^{\frac{1}{3}}} = \frac{1}{2}$$ 7. Final answers: - $$\sqrt[3]{32} = 2 \times 2^{\frac{2}{3}}$$ (or approximately 3.17) - $$\left(\frac{125}{1000}\right)^{\frac{1}{3}} = \frac{1}{2}$$