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.