1. **State the problem:** Simplify the expression $$\left(\frac{100a}{25a^2 b^{\frac{1}{2}}}\right)^{\frac{1}{2}}$$. 2. **Write the formula and rules:** When simplifying expressions with exponents and radicals, use the rules: - $$\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$$ - $$\sqrt{x} = x^{\frac{1}{2}}$$ - $$a^m \div a^n = a^{m-n}$$ - $$\sqrt{a^2} = |a|$$ (absolute value if needed) 3. **Simplify inside the parentheses first:** $$\frac{100a}{25a^2 b^{\frac{1}{2}}} = \frac{100}{25} \cdot \frac{a}{a^2} \cdot \frac{1}{b^{\frac{1}{2}}}$$ 4. **Simplify each part:** $$\frac{100}{25} = 4$$ $$\frac{a}{a^2} = a^{1-2} = a^{-1} = \frac{1}{a}$$ 5. **Rewrite the expression:** $$4 \cdot \frac{1}{a} \cdot b^{-\frac{1}{2}} = \frac{4}{a b^{\frac{1}{2}}}$$ 6. **Apply the outer exponent $$\frac{1}{2}$$:** $$\left(\frac{4}{a b^{\frac{1}{2}}}\right)^{\frac{1}{2}} = \frac{4^{\frac{1}{2}}}{\left(a b^{\frac{1}{2}}\right)^{\frac{1}{2}}}$$ 7. **Simplify numerator:** $$4^{\frac{1}{2}} = \sqrt{4} = 2$$ 8. **Simplify denominator:** $$\left(a b^{\frac{1}{2}}\right)^{\frac{1}{2}} = a^{\frac{1}{2}} \cdot \left(b^{\frac{1}{2}}\right)^{\frac{1}{2}} = a^{\frac{1}{2}} b^{\frac{1}{4}}$$ 9. **Final simplified expression:** $$\frac{2}{a^{\frac{1}{2}} b^{\frac{1}{4}}}$$ **Answer:** $$\boxed{\frac{2}{a^{\frac{1}{2}} b^{\frac{1}{4}}}}$$