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Let's analyze and solve the given equations step-by-step.

### Equations Given:
1. \( A \cdot (B)^4 = B^7 \)
2. \( B(64) = B^7 \)

From equation (2), we have:
\[
B(64) = B^7
\]
This can be rewritten as:
\[
64 \cdot B = B^7
\]
Dividing both sides by \( B \) (assuming \( B \neq 0 \)), we get:
\[
64 = B^6
\]
Taking the 6th root of both sides:
\[
B = \sqrt[6]{64}
\]
\[
B = 2
\]

Now that we have \( B = 2 \), let's substitute this value into equation (1):
\[
A \cdot (2)^4 = 2^7
\]
Simplifying further:
\[
A \cdot 16 = 128
\]
Dividing both sides by 16:
\[
A = \frac{128}{16}
\]
\[
A = 8
\]

### Solution:
- \( A = 8 \)
- \( B = 2 \)

Would you like more details on the solution or any specific questions about this problem? Here are five related questions to explore further:

1. What are the properties of exponents used in the given equations?
2. How would the solution change if \( B \neq 2 \)?
3. Can we solve the problem using logarithms? If so, how?
4. What if the equation had \( A \cdot (B)^n = B^m \)? How would we generalize the solution?
5. How can we verify the solution of \( A \) and \( B \) by substituting back into the original equations?

**Tip:** Always verify the solution by substituting the values back into the original equations to ensure they satisfy both conditions.

The following two equations are true:
A(B)^4 = B^7
B(64) = B^7
What are the values of A and B?

Learn how to solve exponential equations involving variables A and B, with the equations A(B)^4 = B^7 and B(64) = B^7. This problem involves algebraic manipulation and solving systems of equations. Suitable for high school students (Grades 9-12).

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