1. Stating the problem: Simplify the expression $$A= \frac{a^{\frac{1}{5}}\times \sqrt[3]{\sqrt{b}}}{\sqrt[4]{b^3}\times \sqrt{a}}$$. 2. Rewrite radicals using fractional exponents: - $$\sqrt{b} = b^{\frac{1}{2}}$$, so $$\sqrt[3]{\sqrt{b}} = \left(b^{\frac{1}{2}}\right)^{\frac{1}{3}} = b^{\frac{1}{2} \times \frac{1}{3}} = b^{\frac{1}{6}}$$. - $$\sqrt[4]{b^3} = b^{\frac{3}{4}}$$. - $$\sqrt{a} = a^{\frac{1}{2}}$$. 3. Substitute these into the original expression: $$A = \frac{a^{\frac{1}{5}} \times b^{\frac{1}{6}}}{b^{\frac{3}{4}} \times a^{\frac{1}{2}}}$$. 4. Use properties of exponents to combine terms: - For $$a$$ terms: $$a^{\frac{1}{5}} \div a^{\frac{1}{2}} = a^{\frac{1}{5} - \frac{1}{2}} = a^{-\frac{3}{10}}$$. - For $$b$$ terms: $$b^{\frac{1}{6}} \div b^{\frac{3}{4}} = b^{\frac{1}{6} - \frac{3}{4}} = b^{-\frac{7}{12}}$$. 5. Write the expression with negative exponents as positive exponents in the denominator: $$A = \frac{1}{a^{\frac{3}{10}} \times b^{\frac{7}{12}}}$$. Final answer: $$A = \frac{1}{a^{\frac{3}{10}} b^{\frac{7}{12}}}$$.