To simplify the expression $\frac{\sqrt{a} \cdot \sqrt{b}}{\sqrt[3]{ab}}$ using the Law of Exponents, follow these steps:

1. Express square roots and cube roots as exponents:

   • $\sqrt{a} = a^{1/2}$
   • $\sqrt{b} = b^{1/2}$
   • $\sqrt[3]{ab} = (ab)^{1/3}$

   Therefore, the expression becomes:

   $\frac{a^{1/2} \cdot b^{1/2}}{(ab)^{1/3}}$

2. Rewrite the denominator:

   • $(ab)^{1/3}$ can be expressed as $a^{1/3} \cdot b^{1/3}$

   So, the expression now is:

   $\frac{a^{1/2} \cdot b^{1/2}}{a^{1/3} \cdot b^{1/3}}$

3. Apply the Law of Exponents (subtract exponents):

   • For $a$: $\frac{a^{1/2}}{a^{1/3}} = a^{1/2 - 1/3} = a^{3/6 - 2/6} = a^{1/6}$
   • For $b$: $\frac{b^{1/2}}{b^{1/3}} = b^{1/2 - 1/3} = b^{3/6 - 2/6} = b^{1/6}$

   Thus, the simplified expression is:

   $a^{1/6} \cdot b^{1/6}$

4. Combine the terms:

   $a^{1/6} \cdot b^{1/6} = (ab)^{1/6}$

So, the simplified expression is $(ab)^{1/6}$.

Would you like further details or have any questions?

Here are 5 related questions to consider:

1. How would the expression change if we had $\sqrt{a} \cdot \sqrt{b} \cdot \sqrt[3]{a}$ instead?
2. What if the denominator was $\sqrt[3]{a^2b^2}$ instead of $\sqrt[3]{ab}$?
3. How do you simplify $\frac{a^{2/3} \cdot b^{1/4}}{a^{1/2} \cdot b^{1/3}}$?
4. What is the simplified form of $\sqrt[4]{a^2b^3}$?
5. How does the Law of Exponents apply when multiplying expressions with different bases?

Tip: When simplifying expressions involving roots and exponents, always convert the roots to fractional exponents before applying the laws of exponents.