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We are tasked with simplifying the given expression:

\[
25^{x-4} \cdot 5^{3x+1} \cdot 125^{2-x}
\]

Let's break down the problem step by step.

### Step 1: Express everything in terms of base 5
We can express \( 25 \) and \( 125 \) in terms of base \( 5 \):
- \( 25 = 5^2 \)
- \( 125 = 5^3 \)

### Step 2: Rewrite the expression
Now, substitute \( 25 \) and \( 125 \) in terms of base 5:
\[
25^{x-4} = (5^2)^{x-4} = 5^{2(x-4)} = 5^{2x - 8}
\]
\[
125^{2-x} = (5^3)^{2-x} = 5^{3(2-x)} = 5^{6 - 3x}
\]

Thus, the original expression becomes:
\[
5^{2x-8} \cdot 5^{3x+1} \cdot 5^{6-3x}
\]

### Step 3: Combine the exponents
Since all the terms are now in base 5, we can combine them by adding the exponents:
\[
5^{(2x - 8) + (3x + 1) + (6 - 3x)}
\]

Simplify the exponent:
\[
(2x - 8) + (3x + 1) + (6 - 3x) = 2x + 3x - 3x - 8 + 1 + 6 = 2x - 1
\]

Thus, the simplified expression is:
\[
5^{2x - 1}
\]

### Step 4: Match with the options
From the given options, the correct one is:
\[
\boxed{B} \, 5^{2x - 1}
\]

Let me know if you need further details or explanations.

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Here are 5 related questions:
1. How do you simplify expressions with exponents that have different bases?
2. What are the properties of exponents when multiplying terms with the same base?
3. How can we express higher powers in terms of smaller prime bases?
4. What happens when negative exponents appear in a multiplication of terms?
5. How can exponents be simplified when they involve binomials?

**Tip**: Always remember to combine exponents by addition when multiplying terms with the same base.

Өрнекті ықшамдаңыз: 25^(x-4) ⋅ 5^(3x+1) ⋅ 125^(2-x)

Learn how to simplify the exponential expression 25^(x-4) ⋅ 5^(3x+1) ⋅ 125^(2-x) by rewriting terms in base 5. This step-by-step guide covers exponents and their rules, providing a clear solution for students in Grades 9-11.

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