1. **Problem (a):** Express $\sqrt{50} - \frac{2}{\sqrt{8}}$ in the form $a\sqrt{b}$, where $a$ and $b$ are rational numbers. 2. **Step 1:** Simplify each term separately. - $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$ - $\sqrt{8} = \sqrt{2^3} = \sqrt{4 \times 2} = 2\sqrt{2}$ 3. **Step 2:** Rewrite the expression: $$5\sqrt{2} - \frac{2}{2\sqrt{2}}$$ 4. **Step 3:** Simplify the fraction: $$\frac{2}{2\sqrt{2}} = \frac{\cancel{2}}{\cancel{2}\sqrt{2}} = \frac{1}{\sqrt{2}}$$ 5. **Step 4:** Rationalize the denominator of $\frac{1}{\sqrt{2}}$: $$\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$$ 6. **Step 5:** Substitute back: $$5\sqrt{2} - \frac{\sqrt{2}}{2}$$ 7. **Step 6:** Express with common denominator: $$\frac{10\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = \frac{9\sqrt{2}}{2}$$ 8. **Final answer (a):** $$\boxed{\frac{9}{2}\sqrt{2}}$$ --- 1. **Problem (b):** Simplify $\sqrt{\frac{25w^6}{x^{-2}} \div \frac{wx^3}{2}}$, leaving your answer with positive exponents. 2. **Step 1:** Rewrite the division inside the square root as multiplication by reciprocal: $$\sqrt{\frac{25w^6}{x^{-2}} \times \frac{2}{wx^3}}$$ 3. **Step 2:** Multiply numerators and denominators: $$\sqrt{\frac{25w^6 \times 2}{x^{-2} \times wx^3}} = \sqrt{\frac{50w^6}{w x^{-2} x^3}}$$ 4. **Step 3:** Simplify denominator: $$w x^{-2} x^3 = w x^{(-2 + 3)} = w x^{1} = w x$$ 5. **Step 4:** Substitute back: $$\sqrt{\frac{50 w^6}{w x}} = \sqrt{\frac{50 w^6}{w x}}$$ 6. **Step 5:** Simplify fraction inside the root: $$\frac{50 w^6}{w x} = 50 w^{6-1} x^{-1} = 50 w^5 x^{-1}$$ 7. **Step 6:** Write the square root: $$\sqrt{50 w^5 x^{-1}} = \sqrt{50} \times \sqrt{w^5} \times \sqrt{x^{-1}}$$ 8. **Step 7:** Simplify each root: - $\sqrt{50} = 5\sqrt{2}$ - $\sqrt{w^5} = w^{\frac{5}{2}} = w^2 w^{\frac{1}{2}} = w^2 \sqrt{w}$ - $\sqrt{x^{-1}} = x^{-\frac{1}{2}} = \frac{1}{\sqrt{x}}$ 9. **Step 8:** Combine all: $$5\sqrt{2} \times w^2 \sqrt{w} \times \frac{1}{\sqrt{x}} = 5 w^2 \frac{\sqrt{2w}}{\sqrt{x}} = 5 w^2 \sqrt{\frac{2w}{x}}$$ 10. **Final answer (b):** $$\boxed{5 w^2 \sqrt{\frac{2w}{x}}}$$