1. **State the problem:** Simplify the expression $$\left(\frac{100x^4 y^3}{4x^8 y^{-1}}\right)^{\frac{1}{2}}$$ completely and write the answer using positive exponents. 2. **Write the formula and rules:** When simplifying expressions with exponents, use the quotient rule $$\frac{a^m}{a^n} = a^{m-n}$$ and the power of a power rule $$(a^m)^n = a^{mn}$$. 3. **Simplify inside the parentheses first:** $$\frac{100x^4 y^3}{4x^8 y^{-1}} = \frac{100}{4} \cdot \frac{x^4}{x^8} \cdot \frac{y^3}{y^{-1}}$$ 4. **Simplify each part:** - $$\frac{100}{4} = 25$$ - $$\frac{x^4}{x^8} = x^{4-8} = x^{-4}$$ - $$\frac{y^3}{y^{-1}} = y^{3 - (-1)} = y^{4}$$ So the expression inside the parentheses is: $$25 x^{-4} y^{4}$$ 5. **Apply the outer exponent $$\frac{1}{2}$$:** $$\left(25 x^{-4} y^{4}\right)^{\frac{1}{2}} = 25^{\frac{1}{2}} \cdot (x^{-4})^{\frac{1}{2}} \cdot (y^{4})^{\frac{1}{2}}$$ 6. **Simplify each term:** - $$25^{\frac{1}{2}} = 5$$ - $$(x^{-4})^{\frac{1}{2}} = x^{-4 \times \frac{1}{2}} = x^{-2}$$ - $$(y^{4})^{\frac{1}{2}} = y^{4 \times \frac{1}{2}} = y^{2}$$ 7. **Combine the terms:** $$5 x^{-2} y^{2}$$ 8. **Rewrite with positive exponents:** $$5 \frac{y^{2}}{x^{2}}$$ **Final answer:** $$5 \frac{y^{2}}{x^{2}}$$