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 dividing powers with the same base, subtract exponents: $$a^{m} \div a^{n} = a^{m-n}$$. - Negative exponents mean reciprocal: $$x^{-n} = \frac{1}{x^{n}}$$. - Fractional exponents represent roots: $$x^{\frac{1}{2}} = \sqrt{x}$$. - When raising a power to another power, multiply exponents: $$(x^{m})^{n} = x^{mn}$$. 3. **Simplify inside the parentheses:** $$\frac{100a}{25a^{2}b^{\frac{1}{2}}} = \frac{100}{25} \times \frac{a}{a^{2}} \times \frac{1}{b^{\frac{1}{2}}}$$ 4. Simplify each part: $$\frac{100}{25} = 4$$ $$\frac{a}{a^{2}} = a^{1-2} = a^{-1}$$ $$\frac{1}{b^{\frac{1}{2}}} = b^{-\frac{1}{2}}$$ 5. Combine: $$4 \times a^{-1} \times b^{-\frac{1}{2}} = 4a^{-1}b^{-\frac{1}{2}}$$ 6. Now apply the outer exponent $$-\frac{1}{2}$$: $$\left(4a^{-1}b^{-\frac{1}{2}}\right)^{-\frac{1}{2}} = 4^{-\frac{1}{2}} \times a^{-1 \times -\frac{1}{2}} \times b^{-\frac{1}{2} \times -\frac{1}{2}}$$ 7. Calculate each exponent: $$4^{-\frac{1}{2}} = \frac{1}{\sqrt{4}} = \frac{1}{2}$$ $$a^{\frac{1}{2}}$$ $$b^{\frac{1}{4}}$$ 8. Combine all: $$\frac{1}{2} a^{\frac{1}{2}} b^{\frac{1}{4}} = \frac{\sqrt{a} b^{\frac{1}{4}}}{2}$$ **Final answer:** $$\frac{\sqrt{a} b^{\frac{1}{4}}}{2}$$