1. The problem is to simplify and find the value of the expression $$\frac{\sqrt{a \sqrt{a \sqrt{a}}}}{\sqrt[3]{a^{5/2} a}}$$. 2. First, rewrite the expression inside the radicals using exponent rules. 3. Simplify the numerator: $$\sqrt{a \sqrt{a \sqrt{a}}} = \sqrt{a \cdot a^{1/2} \cdot a^{1/4}} = \sqrt{a^{1 + \frac{1}{2} + \frac{1}{4}}} = \sqrt{a^{\frac{7}{4}}} = a^{\frac{7}{8}}$$ 4. Simplify the denominator: $$\sqrt[3]{a^{5/2} a} = \sqrt[3]{a^{\frac{5}{2} + 1}} = \sqrt[3]{a^{\frac{7}{2}}} = a^{\frac{7}{6}}$$ 5. Now the expression is: $$\frac{a^{\frac{7}{8}}}{a^{\frac{7}{6}}} = a^{\frac{7}{8} - \frac{7}{6}}$$ 6. Find the exponent difference: $$\frac{7}{8} - \frac{7}{6} = \frac{21}{24} - \frac{28}{24} = -\frac{7}{24}$$ 7. So the expression simplifies to: $$a^{-\frac{7}{24}} = \frac{1}{a^{\frac{7}{24}}}$$ Final answer: $$\boxed{\frac{1}{a^{\frac{7}{24}}}}$$