1. **Stating the problem:** Simplify the expression $$N = \frac{\sqrt[3]{a^{3} \sqrt{a^{2} \sqrt{a^{5} \sqrt{a^{6}}}}}}{\sqrt[3]{\sqrt{a^{22}}}}$$. 2. **Rewrite nested radicals as exponents:** Recall that $$\sqrt[n]{a^{m}} = a^{\frac{m}{n}}$$. 3. Simplify the innermost radical in the numerator: $$\sqrt{a^{6}} = a^{\frac{6}{2}} = a^{3}$$. 4. Substitute back: $$\sqrt{a^{5} \sqrt{a^{6}}} = \sqrt{a^{5} \cdot a^{3}} = \sqrt{a^{8}} = a^{\frac{8}{2}} = a^{4}$$. 5. Next: $$\sqrt{a^{2} \sqrt{a^{5} \sqrt{a^{6}}}} = \sqrt{a^{2} \cdot a^{4}} = \sqrt{a^{6}} = a^{\frac{6}{2}} = a^{3}$$. 6. Now the numerator inside the cube root is: $$a^{3} \sqrt{a^{2} \sqrt{a^{5} \sqrt{a^{6}}}} = a^{3} \cdot a^{3} = a^{6}$$. 7. So numerator becomes: $$\sqrt[3]{a^{6}} = a^{\frac{6}{3}} = a^{2}$$. 8. Simplify the denominator: $$\sqrt{a^{22}} = a^{\frac{22}{2}} = a^{11}$$. 9. Then: $$\sqrt[3]{\sqrt{a^{22}}} = \sqrt[3]{a^{11}} = a^{\frac{11}{3}}$$. 10. Now the entire expression is: $$N = \frac{a^{2}}{a^{\frac{11}{3}}} = a^{2 - \frac{11}{3}} = a^{\frac{6}{3} - \frac{11}{3}} = a^{-\frac{5}{3}}$$. 11. Final simplified form: $$N = a^{-\frac{5}{3}} = \frac{1}{a^{\frac{5}{3}}}$$. **Answer:** $$\boxed{\frac{1}{a^{\frac{5}{3}}}}$$