1. **State the problem:** Simplify the expression $$\frac{2ab^{-1}}{2^{-1}a^{-4}b^2}$$ and write the result using positive exponents only. 2. **Recall the rules:** - When dividing powers with the same base, subtract exponents: $$\frac{x^m}{x^n} = x^{m-n}$$. - Negative exponents mean reciprocal: $$x^{-m} = \frac{1}{x^m}$$. - Simplify coefficients separately. 3. **Rewrite the expression:** $$\frac{2ab^{-1}}{2^{-1}a^{-4}b^2} = \frac{2 \cdot a \cdot b^{-1}}{2^{-1} \cdot a^{-4} \cdot b^2}$$ 4. **Divide coefficients:** $$\frac{2}{2^{-1}} = 2 \times 2^{1} = 2^1 \times 2^1 = 2^{1+1} = 2^2 = 4$$ 5. **Divide variables with the same base by subtracting exponents:** - For $a$: $$a^{1} \div a^{-4} = a^{1 - (-4)} = a^{1+4} = a^{5}$$ - For $b$: $$b^{-1} \div b^{2} = b^{-1 - 2} = b^{-3}$$ 6. **Combine all parts:** $$4 \cdot a^{5} \cdot b^{-3}$$ 7. **Rewrite with positive exponents only:** $$4a^{5} \frac{1}{b^{3}} = \frac{4a^{5}}{b^{3}}$$ **Final answer:** $$\frac{4a^{5}}{b^{3}}$$