1. **State the problem:** Simplify the expression $$\left(\frac{100a}{25a^{2}b^{\frac{1}{2}}}\right)^{-\frac{1}{2}}$$. 2. **Recall the rules:** - When raising a fraction to a power, apply the power to numerator and denominator. - Negative exponents mean reciprocal: $$x^{-n} = \frac{1}{x^n}$$. - When dividing powers with the same base, subtract exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. - Simplify radicals and fractional exponents carefully. 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: $$\frac{100}{25} = 4$$ $$\frac{a}{a^{2}} = a^{1-2} = a^{-1}$$ 5. So inside the parentheses is: $$4 \cdot a^{-1} \cdot b^{-\frac{1}{2}} = 4a^{-1}b^{-\frac{1}{2}}$$ 6. Now apply the exponent $$-\frac{1}{2}$$ to the entire expression: $$\left(4a^{-1}b^{-\frac{1}{2}}\right)^{-\frac{1}{2}} = 4^{-\frac{1}{2}} \cdot a^{-1 \cdot -\frac{1}{2}} \cdot b^{-\frac{1}{2} \cdot -\frac{1}{2}}$$ 7. Simplify each term: $$4^{-\frac{1}{2}} = \frac{1}{\sqrt{4}} = \frac{1}{2}$$ $$a^{\frac{1}{2}}$$ $$b^{\frac{1}{4}}$$ 8. Combine all: $$\frac{1}{2} a^{\frac{1}{2}} b^{\frac{1}{4}}$$ **Final answer:** $$\frac{1}{2} a^{\frac{1}{2}} b^{\frac{1}{4}}$$