Let's solve the given expression step by step.

The expression provided is:

$\frac{3^{-2}}{3^{-3} \cdot 3^0}$

Step 1: Simplify the denominator

We have the product of two powers of 3 in the denominator:

$3^{-3} \cdot 3^0$

Using the property of exponents $a^m \cdot a^n = a^{m+n}$, we combine the powers of 3:

$3^{-3} \cdot 3^0 = 3^{-3+0} = 3^{-3}$

Step 2: Rewrite the expression

Now the entire expression becomes:

$\frac{3^{-2}}{3^{-3}}$

Step 3: Use the quotient of powers property

Using the property $\frac{a^m}{a^n} = a^{m-n}$, we simplify the expression:

$\frac{3^{-2}}{3^{-3}} = 3^{-2 - (-3)} = 3^{-2 + 3} = 3^1$

Step 4: Final result

The expression simplifies to:

$3^1 = 3$

Thus, the equivalent expression is 3.

Would you like more details or have any questions? Here are some related questions to deepen understanding:

1. What is the general rule for multiplying powers with the same base?
2. How would the result change if the exponents were positive instead of negative?
3. Can you explain why $3^0 = 1$ and how it affects the denominator?
4. How would the result be affected if a different base were used?
5. How can the properties of exponents help in solving more complex expressions?

Tip: Always simplify powers using the exponent rules before applying other operations, as it makes complex expressions easier to handle.