The given expression is:

\[
\frac{3^3 \cdot 3^{-1}}{3^5 \cdot 3^2}
\]

We can simplify this step by step using the properties of exponents. Recall that:

1. \( a^m \cdot a^n = a^{m+n} \)
2. \( \frac{a^m}{a^n} = a^{m-n} \)

### Step 1: Simplify the numerator
The numerator is \( 3^3 \cdot 3^{-1} \). Using the property of exponents:

\[
3^3 \cdot 3^{-1} = 3^{3+(-1)} = 3^2
\]

### Step 2: Simplify the denominator
The denominator is \( 3^5 \cdot 3^2 \). Using the same property of exponents:

\[
3^5 \cdot 3^2 = 3^{5+2} = 3^7
\]

### Step 3: Combine the numerator and denominator
Now, we have:

\[
\frac{3^2}{3^7}
\]

Using the division property of exponents:

\[
\frac{3^2}{3^7} = 3^{2-7} = 3^{-5}
\]

### Step 4: Simplify the final expression
\[
3^{-5} = \frac{1}{3^5}
\]

Thus, the simplified value of the expression is:

\[
\frac{1}{243}
\]

Would you like more details or further explanations?

Here are 5 related questions to expand your understanding:
1. How do you multiply two numbers with the same base but different exponents?
2. What happens when you raise a number to a negative exponent?
3. How do you divide numbers with the same base and different exponents?
4. How would you simplify the expression \( \frac{a^3 \cdot a^{-2}}{a^4 \cdot a^3} \)?
5. What is the value of \( 2^3 \cdot 2^{-4} \)?

**Tip:** Always remember to add exponents when multiplying powers of the same base and subtract them when dividing.

Learn how to simplify the exponential expression 3^3 * 3^(-1) / 3^5 * 3^2 with a step-by-step solution. This problem covers key concepts in exponents, including multiplying and dividing powers with the same base, and is suitable for students in Grades 8-10.

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