The given expression is:

$\frac{\sqrt{a} \cdot \sqrt{b}}{\sqrt[9]{ab}}$

We can simplify this using the properties of exponents and radicals. Let's proceed step by step.

Step 1: Express radicals in exponential form

Recall that:

• $\sqrt{a} = a^{1/2}$
• $\sqrt{b} = b^{1/2}$
• $\sqrt[9]{ab} = (ab)^{1/9}$

Thus, the expression becomes: $\frac{a^{1/2} \cdot b^{1/2}}{(ab)^{1/9}}$

Step 2: Simplify the denominator

Since $(ab)^{1/9} = a^{1/9} \cdot b^{1/9}$, the expression becomes: $\frac{a^{1/2} \cdot b^{1/2}}{a^{1/9} \cdot b^{1/9}}$

Step 3: Apply the laws of exponents

For division, subtract the exponents: $a^{1/2 - 1/9} \cdot b^{1/2 - 1/9}$

Step 4: Simplify the exponents

• $1/2 - 1/9 = \frac{9}{18} - \frac{2}{18} = \frac{7}{18}$

Thus, the simplified expression becomes: $a^{7/18} \cdot b^{7/18}$

Final Answer:

$(a \cdot b)^{7/18}$

Let me know if you need further details or have any questions! Here are some related problems to explore:

1. Simplify $\frac{\sqrt[3]{x} \cdot \sqrt{y}}{\sqrt[6]{xy}}$.
2. Simplify $\frac{a^{1/3} \cdot b^{1/4}}{a^{1/5} \cdot b^{1/6}}$.
3. What is $\sqrt[5]{x^3 \cdot y^2}$ in exponential form?
4. Combine and simplify $\frac{x^{2/3} \cdot y^{4/3}}{x^{1/3} \cdot y^{2/3}}$.
5. Simplify $\frac{x^{1/4} \cdot x^{1/2}}{\sqrt{x}}$.

Tip: When simplifying radicals and exponents, always convert them to their exponential form to make it easier to apply exponent rules.