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We are given the following system of equations:

\[
2^{x + y} = 125 \quad \text{and} \quad 2^{x - y} = 25
\]

We are asked to determine the value of \( 2^{\frac{y}{x}} \) in closed form.

### Step 1: Expressing the equations with logarithms
Start by rewriting the given powers in terms of logarithms. First, express 125 and 25 as powers of 5:

\[
125 = 5^3 \quad \text{and} \quad 25 = 5^2
\]

Thus, we rewrite the system as:

\[
2^{x + y} = 5^3 \quad \text{and} \quad 2^{x - y} = 5^2
\]

### Step 2: Taking logarithms on both sides
Take the logarithm of both sides of each equation. Using natural logarithms (or base 2 logs) is appropriate, but we proceed generally:

For the first equation:
\[
x + y = \log_2(5^3) = 3\log_2(5)
\]

For the second equation:
\[
x - y = \log_2(5^2) = 2\log_2(5)
\]

### Step 3: Solving the system of linear equations
We now have the following system of linear equations:

\[
x + y = 3\log_2(5)
\]
\[
x - y = 2\log_2(5)
\]

Add the two equations to eliminate \( y \):

\[
2x = 5\log_2(5)
\]

Thus,
\[
x = \frac{5\log_2(5)}{2}
\]

Now, subtract the second equation from the first to eliminate \( x \):

\[
2y = \log_2(5)
\]

Thus,
\[
y = \frac{\log_2(5)}{2}
\]

### Step 4: Finding \( 2^{\frac{y}{x}} \)
We want to find \( 2^{\frac{y}{x}} \). From the expressions for \( x \) and \( y \), we have:

\[
\frac{y}{x} = \frac{\frac{\log_2(5)}{2}}{\frac{5\log_2(5)}{2}} = \frac{1}{5}
\]

Therefore:
\[
2^{\frac{y}{x}} = 2^{\frac{1}{5}}
\]

Thus, the value of \( 2^{\frac{y}{x}} \) in closed form is \( 2^{\frac{1}{5}} \).

Would you like further details or clarifications?

Here are 5 related questions:
1. How would you solve this if the numbers were different powers of a different base, like 3 or 7?
2. What is the significance of taking logarithms in solving exponential equations?
3. Can you find a general method for solving equations of the form \( a^{x+y} = m \) and \( a^{x-y} = n \)?
4. How does this approach change if the system involved more variables, such as \( x+y+z \)?
5. What other mathematical techniques can be used for solving nonlinear systems involving exponentials?

**Tip**: When solving exponential equations, expressing the constants as powers of the same base can greatly simplify the solution process.

Suppose 2^(x+y) = 125 and 2^(x−y) = 25, without using a calculator determine in closed form, the value of 2^(y/x).

This problem demonstrates how to solve the system of exponential equations 2^(x+y) = 125 and 2^(x−y) = 25 to determine the value of 2^(y/x) in closed form, using logarithms and properties of exponents. Suitable for advanced high school or undergraduate students.

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