https://file.aiquickdraw.com/mathbot/file/1726797451232f4ty40td.jpg

Let's break down and solve the two questions provided in the image:

### First Question:
The statements involve exponentiation:
1. \( 5^3 = 5 \times 5 \times 5 \) (This is true, as \( 5^3 = 125 \)).
2. \( 3^6 = 6 \times 6 \times 6 \) (This is **false**; \( 3^6 \) is not equal to multiplying 6 six times. The correct expansion would be \( 3 \times 3 \times 3 \times 3 \times 3 \times 3 \)).
3. \( (12)^4 = 12 \times 12 \times 12 \times 12 \) (This is true, as \( 12^4 \) involves multiplying 12 four times).
4. \( 2^5 = 5 \times 5 \) (This is **false**; \( 2^5 \) equals \( 2 \times 2 \times 2 \times 2 \times 2 \), not \( 5 \times 5 \)).

Thus, the correct statements are **(1) and (3)**.

### Second Question:
We are given the expression:
\[
\frac{a^2 \cdot b^3}{c^4}
\]
Where \( a = 4 \), \( b = 3 \), and \( c = 2 \). Substituting the values:

\[
\frac{4^2 \cdot 3^3}{2^4} = \frac{16 \cdot 27}{16}
\]

Simplifying:

\[
\frac{432}{16} = 27
\]

Thus, the correct answer is **27**.

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Would you like further details or have any other questions?

### Additional Related Questions:
1. What is the expanded form of \( (5^4) \)?
2. Can you explain why \( 2^5 \neq 5 \times 5 \)?
3. How would you simplify \( \frac{(a^3 \cdot b^4)}{c^2} \) with similar values?
4. What is the result of \( 7^2 \times 3^2 \)?
5. How does exponentiation work with negative bases?

### Tip:
When dealing with exponents, remember that the exponent represents how many times the base is multiplied by itself!

1. (1) 5^3 = 5 × 5 × 5 (True), (2) 3^6 = 6 × 6 × 6 (False), (3) (12)^4 = 12 × 12 × 12 × 12 (True), (4) 2^5 = 5 × 5 (False). Which statements are correct? 2. If a = 4, b = 3, and c = 2, find the value of (a^2 * b^3) / c^4.

Analyze statements involving exponentiation, including determining correctness and solving expressions like (a^2 * b^3) / c^4. This problem is suitable for students in Grades 7-9, covering key exponent rules.

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