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The question asks for the value of the expression:

\[
\left( 3\sqrt{3} \right)^{-2}
\]

Let's solve this step by step:

### Step 1: Express the square root as an exponent
The square root of 3, \(\sqrt{3}\), can be rewritten as \(3^{1/2}\). So the given expression becomes:

\[
\left( 3 \cdot 3^{1/2} \right)^{-2}
\]

### Step 2: Simplify the base
Now simplify the base using exponent rules. We combine the exponents of 3:

\[
3^{1} \cdot 3^{1/2} = 3^{1 + 1/2} = 3^{3/2}
\]

### Step 3: Apply the power of \(-2\)
Now raise the expression to the power of \(-2\):

\[
\left( 3^{3/2} \right)^{-2} = 3^{(3/2) \cdot (-2)} = 3^{-3}
\]

### Step 4: Simplify the result
We know that \(3^{-3} = \frac{1}{3^3} = \frac{1}{27}\).

### Final Answer:
The value of \(\left( 3\sqrt{3} \right)^{-2}\) is \(\frac{1}{27}\).

So the correct answer is **(C) \( \frac{1}{27} \)**.

Would you like further details or have any questions?

Here are some related questions to deepen your understanding:
1. What is the general rule for multiplying exponents with the same base?
2. How would the result change if the exponent were positive instead of negative?
3. How can you express \( \sqrt{5} \) using exponents?
4. What happens if you raise a number to a fractional exponent like \( 3^{1/3} \)?
5. Can you simplify \( (a^m \cdot a^n)^p \) in a general form?

**Tip:** When dealing with exponents, always try to simplify the base first, especially when you have terms like square roots or cube roots, before applying any power.

Learn how to solve the expression (3√3)^-2 using exponent and square root rules. This problem introduces algebraic manipulation of exponents and fractional powers, suitable for students in Grades 9-10.

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