Let's solve the expression step by step:

\[
(3x)^2 \times \left( \frac{1}{9x} \right)^{\frac{1}{2}}
\]

### Step 1: Simplify \((3x)^2\)
\[
(3x)^2 = 9x^2
\]

### Step 2: Simplify \(\left( \frac{1}{9x} \right)^{\frac{1}{2}}\)
\[
\left( \frac{1}{9x} \right)^{\frac{1}{2}} = \frac{1}{\sqrt{9x}} = \frac{1}{3\sqrt{x}}
\]

### Step 3: Multiply the two expressions
\[
9x^2 \times \frac{1}{3\sqrt{x}} = \frac{9x^2}{3\sqrt{x}} = 3x^{2} \div \sqrt{x}
\]

### Step 4: Simplify \(\frac{x^2}{\sqrt{x}}\)
\[
\frac{x^2}{\sqrt{x}} = x^{2 - \frac{1}{2}} = x^{\frac{3}{2}}
\]

### Final expression:
\[
3x^{\frac{3}{2}}
\]

So, the simplified form of the expression is:
\[
3x^{\frac{3}{2}}
\]

Would you like further clarification or more details on any of these steps?

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Here are 5 related questions to deepen your understanding:

1. How do you simplify expressions with fractional exponents?
2. What is the process for multiplying terms with exponents?
3. How do you simplify radical expressions involving variables?
4. How does raising a fraction to a power work in different cases?
5. What is the relationship between exponents and roots?

**Tip:** When simplifying expressions with exponents, always pay attention to the exponent rules, especially when handling radicals and fractional exponents.

Learn how to simplify the expression (3x)^2 * (1/9x)^(1/2) with a detailed, step-by-step breakdown. This problem involves key algebraic concepts like exponentiation, radical simplification, and multiplication of terms with exponents. Suitable for students in Grades 9-11.

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