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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.

Using laws of exponents, simplify the expression \(\frac{\sqrt{a} \sqrt{b}}{\sqrt[9]{ab}}\).

Learn how to simplify the radical expression \(\frac{\sqrt{a} \sqrt{b}}{\sqrt[9]{ab}}\) using the laws of exponents and radicals. This step-by-step solution breaks down the problem by converting radicals into exponents and applying exponent subtraction. Suitable for students in Grades 9-12.

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