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The given mathematical expression to simplify is:

\[
\left( \frac{9a^2 b^{-1} c}{27a^3 b^2 c^2} \right)^{-1}
\]

### Step 1: Simplify the fraction inside the parentheses

We begin by simplifying the fraction:

\[
\frac{9a^2 b^{-1} c}{27a^3 b^2 c^2}
\]

This can be broken down as follows:
- For the constants: \( \frac{9}{27} = \frac{1}{3} \)
- For the powers of \( a \): \( \frac{a^2}{a^3} = a^{2-3} = a^{-1} \)
- For the powers of \( b \): \( \frac{b^{-1}}{b^2} = b^{-1-2} = b^{-3} \)
- For the powers of \( c \): \( \frac{c}{c^2} = c^{1-2} = c^{-1} \)

Thus, the fraction simplifies to:

\[
\frac{1}{3} a^{-1} b^{-3} c^{-1}
\]

### Step 2: Apply the negative exponent

The expression now becomes:

\[
\left( \frac{1}{3} a^{-1} b^{-3} c^{-1} \right)^{-1}
\]

Using the property of exponents, \( (x^{-1}) = \frac{1}{x} \), the expression becomes:

\[
3 a^1 b^3 c^1 = 3ab^3c
\]

### Final Answer:
The simplified form is \( 3ab^3c \), which matches option **A**.

Let me know if you need more details or further clarification!

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Here are 5 related questions for further practice:
1. How do you simplify expressions with negative exponents?
2. What is the rule for dividing terms with the same base but different exponents?
3. How does multiplying by the reciprocal work when dealing with negative exponents?
4. Can you simplify \( \frac{4x^{-2}}{2x^3} \)?
5. What happens when you apply a power of negative one to a fraction?

**Tip:** Always combine like terms (same base) by adding or subtracting exponents when multiplying or dividing powers.

Bentuk sederhana dari \left( \frac{9a^2 b^{-1} c}{27a^3 b^2 c^2} \right)^{-1} adalah....

Learn how to simplify algebraic expressions with exponents by simplifying \left( \frac{9a^2 b^{-1} c}{27a^3 b^2 c^2} \right)^{-1}. This example covers key algebraic concepts such as exponent rules and fraction simplification, suitable for students in Grades 10-12.

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