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The given expression is:

\[
\sqrt{25 m^{4 - 2x} n^{8y - 6}}
\]

Let's simplify this step by step:

### Step 1: Simplify the square root of 25
\[
\sqrt{25} = 5
\]
So, we have:
\[
5 \sqrt{m^{4 - 2x} n^{8y - 6}}
\]

### Step 2: Apply the square root rule to exponents
For any \(a^b\), we have:
\[
\sqrt{a^b} = a^{b/2}
\]
Now apply this to both \(m^{4 - 2x}\) and \(n^{8y - 6}\):

\[
m^{(4 - 2x)/2} = m^{2 - x}
\]
\[
n^{(8y - 6)/2} = n^{4y - 3}
\]

### Step 3: Combine the terms
Now, we combine everything to get the final simplified expression:
\[
5 m^{2 - x} n^{4y - 3}
\]

This is the simplified form of the expression.

Would you like further details on any step or have any questions?

Here are some related questions for further exploration:
1. What is the general rule for simplifying square roots of exponents?
2. How do we handle expressions with negative exponents under a square root?
3. What is the relationship between exponents and roots in more complex algebraic terms?
4. Can we use these simplifications for cube roots or higher-order roots?
5. How would this expression behave if \(x = 2\) or \(y = 1\)?

**Tip**: When simplifying expressions with square roots, always remember to break down constants and use exponent rules carefully to avoid mistakes.

Simplify the square root expression \( \sqrt{25 m^{4 - 2x} n^{8y - 6}} \)

Learn how to simplify the square root expression \( \sqrt{25 m^{4 - 2x} n^{8y - 6}} \) step by step. This solution covers key algebra concepts, such as exponent rules, and applies them to simplify complex square root expressions. Suitable for high school students.

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