[b] Simplify to the simplest form : \( \sqrt{125}-\sqrt[3]{250}+\frac{1}{2} \sqrt[3]{16}+\sqrt{20} \)

Let's simplify each term step by step: 1. **For \( \sqrt{125} \)**: \[ \sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \cdot \sqrt{5} = 5\sqrt{5} \] 2. **For \( \sqrt[3]{250} \)**: \[ \sqrt[3]{250} = \sqrt[3]{125 \times 2} = \sqrt[3]{125} \cdot \sqrt[3]{2} = 5\sqrt[3]{2} \] 3. **For \( \frac{1}{2} \sqrt[3]{16} \)**: \[ \sqrt[3]{16} = \sqrt[3]{2^4} = 2^{\frac{4}{3}} = 2 \cdot \sqrt[3]{2} \quad (\text{since } 2^{\frac{4}{3}} = 2^{1} \cdot 2^{\frac{1}{3}} = 2\sqrt[3]{2}) \] Thus, \[ \frac{1}{2} \sqrt[3]{16} = \frac{1}{2} \cdot 2 \cdot \sqrt[3]{2} = \sqrt[3]{2} \] 4. **For \( \sqrt{20} \)**: \[ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \] Now we can combine all the simplified terms: \[ \sqrt{125} - \sqrt[3]{250} + \frac{1}{2} \sqrt[3]{16} + \sqrt{20} = 5\sqrt{5} - 5\sqrt[3]{2} + \sqrt[3]{2} + 2\sqrt{5} \] Combine like terms: \[ (5\sqrt{5} + 2\sqrt{5}) + (-5\sqrt[3]{2} + \sqrt[3]{2}) = 7\sqrt{5} - 4\sqrt[3]{2} \] Thus, the expression simplified to its simplest form is: \[ \boxed{7\sqrt{5} - 4\sqrt[3]{2}} \]