https://file.aiquickdraw.com/mathbot/file/1726889607282wexsvgk5.jpg

The image shows the following expression that needs to be simplified:

\[
\frac{12^3 \cdot 3^2}{3^7}
\]

Let's work through the simplification step-by-step.

### Step 1: Simplify the numerator
The expression in the numerator is \( 12^3 \cdot 3^2 \).

Since \( 12 = 3 \times 4 \), we can express \( 12^3 \) as:

\[
12^3 = (3 \times 4)^3 = 3^3 \times 4^3
\]

So, the numerator becomes:

\[
(3^3 \times 4^3) \cdot 3^2 = 3^5 \times 4^3
\]

### Step 2: Combine with the denominator
Now, the full expression becomes:

\[
\frac{3^5 \times 4^3}{3^7}
\]

We can simplify the powers of 3 by subtracting the exponents:

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

Thus, the entire expression simplifies to:

\[
4^3 \cdot 3^{-2} = \frac{4^3}{3^2}
\]

### Step 3: Evaluate the powers
Now, compute the powers:

\[
4^3 = 64 \quad \text{and} \quad 3^2 = 9
\]

So, the final simplified expression is:

\[
\frac{64}{9}
\]

Thus, the simplified form of the expression is \( \frac{64}{9} \).

Would you like more details or have any questions about this process?

Here are 5 related questions you might explore:
1. How do you handle exponents when multiplying terms with the same base?
2. What are the steps for simplifying a fraction with exponents in both the numerator and denominator?
3. How can prime factorization help in simplifying large numbers?
4. What happens when you have negative exponents in an expression?
5. Can you apply these simplification techniques to expressions with variables?

**Tip:** When simplifying expressions with powers, always look for common bases to combine or reduce terms effectively!

Perform the indicated operation and simplify if necessary.

Discover how to simplify expressions using exponentiation and algebraic rules, illustrated through a step-by-step guide. This problem involves simplifying powers in both the numerator and denominator to achieve the final expression.

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