1. **Simplify** $ (8m^{3})^{\frac{1}{3}} $ with positive powers. Step 1: Apply the power of a power rule: $$ (a^{m})^{n} = a^{m \times n} $$ Step 2: Write $$ (8m^{3})^{\frac{1}{3}} = 8^{\frac{1}{3}} \times (m^{3})^{\frac{1}{3}} $$ Step 3: Evaluate: $$ 8^{\frac{1}{3}} = 2 $$ because 2³ = 8. Step 4: Evaluate powers of $$ m $$: $$ (m^{3})^{\frac{1}{3}} = m^{3 \times \frac{1}{3}} = m^{1} = m $$ Step 5: Combine results: $$ (8m^{3})^{\frac{1}{3}} = 2m $$ 2. **Simplify** $ 81^{-\frac{1}{2}} $ with positive powers. Step 1: Write $$ 81^{-\frac{1}{2}} = \frac{1}{81^{\frac{1}{2}}} $$ Step 2: Recognize that $$ 81^{\frac{1}{2}} $$ is the square root of 81. Step 3: Calculate $$ \sqrt{81} = 9 $$ Step 4: Substitute back: $$ 81^{-\frac{1}{2}} = \frac{1}{9} $$ 3. **Calculate the wages bill for 30 men if 10 men’s wages are 1000.** Step 1: Find the wage per man: $$ \text{Wage per man} = \frac{1000}{10} = 100 $$ Step 2: Calculate wages for 30 men: $$ \text{Wages for 30 men} = 30 \times 100 = 3000 $$ **Final answers:** (a) $$ 2m $$ (b) $$ \frac{1}{9} $$ Wages bill for 30 men: $$ 3000 $$