https://r2.erweima.ai/mathbot/image/1726966743608xt4etr4p.jpg

The image contains a mathematical expression being simplified step by step. Here's the main structure of the expression and its simplification:

### Original Expression:
\[
\frac{5 \cdot (3^2 - 10)^2}{3^2 \cdot 60^2}
\]

### Step-by-Step Simplification:

1. **First Line:**
   \[
   \frac{5 \cdot (3^2 - 10)^2}{3^2 \cdot 60^2}
   \]
   Expand \( 3^2 = 9 \) and \( 60^2 = 3600 \):
   \[
   \frac{5 \cdot (9 - 10)^2}{9 \cdot 3600}
   \]

2. **Second Line:**
   \[
   \frac{5 \cdot (-1)^2}{9 \cdot 3600}
   \]
   Since \( (-1)^2 = 1 \):
   \[
   \frac{5 \cdot 1}{9 \cdot 3600}
   \]

3. **Third Line:**
   Simplify the denominator:
   \[
   \frac{5}{32400}
   \]

4. **Fourth Line:**
   Simplify the fraction:
   \[
   \frac{5}{32400} = \frac{5}{(2^2 \cdot 5^2 \cdot 3^4)} = \frac{1}{6480}
   \]

Finally, this expression simplifies further into:
\[
\frac{5}{4}
\]

Thus, the final result is \( \frac{5}{4} \).

Would you like further clarification on any step? Or perhaps you want more details?

### Relative Questions:
1. How do you handle powers of negative numbers in equations?
2. What happens when simplifying complex fractions involving large numbers?
3. Can you explain how to simplify \( (3^2 - 10)^2 \) further in this example?
4. How does factorization help in simplifying the fraction \( \frac{5}{32400} \)?
5. Why is it useful to break down numbers like 60 and 3600 into their prime factors?

### Tip:
Always ensure that powers and signs (positive/negative) are correctly handled during simplifications, as mistakes in these areas are common but crucial to avoid!

Based on the image, provide a structured breakdown of the mathematical simplification process.

Explore the step-by-step simplification of an algebraic expression using exponent laws, highlighting key concepts such as exponentiation and algebraic manipulation. Suitable for high school students learning advanced algebra.

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