https://file.aiquickdraw.com/mathbot/file/1730035609188r68ecs7g.jpg

Let's analyze the question from the image:

The question asks us to find the value of \( a \cdot b \) under these conditions:
1. The value of \( a \) is given.
2. \( b \) satisfies the equation \( 2^{a - 3} \cdot 7^b = 28^a \).

We need to solve for \( a \) and \( b \), then calculate \( a \cdot b \).

### Solution Steps
1. **Rewrite \( 28^a \) in terms of prime factors**:
   - Since \( 28 = 2^2 \cdot 7 \), we can rewrite \( 28^a \) as:
     \[
     28^a = (2^2 \cdot 7)^a = 2^{2a} \cdot 7^a
     \]

2. **Set up the equation**:
   \[
   2^{a - 3} \cdot 7^b = 2^{2a} \cdot 7^a
   \]

3. **Equate the powers of 2 and 7 separately**:
   - For the powers of 2:
     \[
     a - 3 = 2a
     \]
     Solving for \( a \), we get:
     \[
     -3 = a \quad \Rightarrow \quad a = -3
     \]

   - For the powers of 7:
     \[
     b = a \quad \Rightarrow \quad b = -3
     \]

4. **Calculate \( a \cdot b \)**:
   \[
   a \cdot b = (-3) \cdot (-3) = 9
   \]

### Answer
The correct answer is \( 9 \).

Would you like further details on any of these steps, or do you have any questions?

### Follow-up Questions
1. How do we handle equations with different bases in exponentials?
2. What is the importance of equating powers when dealing with equations of this type?
3. Can we solve similar problems using logarithmic properties?
4. Why did we separate the powers of \( 2 \) and \( 7 \) in this solution?
5. What if the base values were not prime factors?

### Tip
Always try to simplify expressions by breaking down numbers into their prime factors for easier handling of exponents.

Given that a and b satisfy the equation 2^(a - 3) * 7^b = 28^a, find the value of a * b.

Learn how to solve the exponential equation 2^(a - 3) * 7^b = 28^a to find the values of a and b, then calculate a * b. This solution includes steps on equating powers and prime factorization. Suitable for students in Grades 10-12.

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