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Let's solve the given system of equations:

\[
\begin{cases}
5^{x+y} = 125, \\
4^{(x-y)^2 - 1} = 1.
\end{cases}
\]

### Step 1: Solve the first equation
The first equation is:
\[
5^{x+y} = 125.
\]
Since \(125 = 5^3\), we can write:
\[
5^{x+y} = 5^3.
\]
Equating the exponents (since the bases are the same), we get:
\[
x + y = 3.
\]

### Step 2: Solve the second equation
The second equation is:
\[
4^{(x-y)^2 - 1} = 1.
\]
We know that any non-zero number raised to the power of 0 is 1, so:
\[
(x-y)^2 - 1 = 0.
\]
Solving for \((x-y)^2\), we get:
\[
(x-y)^2 = 1.
\]
Taking the square root of both sides, we get two possible solutions:
\[
x - y = 1 \quad \text{or} \quad x - y = -1.
\]

### Step 3: Solve the system
Now we have two cases to consider:

**Case 1: \(x - y = 1\)**
- From \(x + y = 3\) and \(x - y = 1\):
\[
\begin{cases}
x + y = 3, \\
x - y = 1.
\end{cases}
\]
Adding these two equations:
\[
2x = 4 \quad \Rightarrow \quad x = 2.
\]
Substituting \(x = 2\) into \(x + y = 3\):
\[
2 + y = 3 \quad \Rightarrow \quad y = 1.
\]

**Case 2: \(x - y = -1\)**
- From \(x + y = 3\) and \(x - y = -1\):
\[
\begin{cases}
x + y = 3, \\
x - y = -1.
\end{cases}
\]
Adding these two equations:
\[
2x = 2 \quad \Rightarrow \quad x = 1.
\]
Substituting \(x = 1\) into \(x + y = 3\):
\[
1 + y = 3 \quad \Rightarrow \quad y = 2.
\]

### Final Answer:
The solutions to the system are:
\[
(x, y) = (2, 1) \quad \text{or} \quad (1, 2).
\]

Do you want more details or have any questions?

Here are five related questions for further exploration:
1. How would the solutions change if the first equation was \(5^{x+y} = 625\)?
2. What if the second equation was \(4^{(x-y)^2 - 2} = 1\)? How would that affect the solutions?
3. Can you solve a similar system where the base of the first equation is not a prime number?
4. What if the exponents were fractional instead of integer values?
5. How does the method change if the equations involve logarithms instead of exponents?

**Tip:** When solving systems involving exponents, it helps to express both sides of the equation with the same base whenever possible.

Learn how to solve the system of exponential equations 5^(x+y) = 125 and 4^((x-y)^2 - 1) = 1 with a detailed step-by-step solution. This problem involves exponential equations and systems of equations solving methods. Suitable for students in Grades 10-12.

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