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To determine which option is equivalent to \(5^{a + 2b}\), we can simplify this expression.

Given:
\[
5^{a + 2b}
\]

Using the property of exponents that states \(x^{m+n} = x^m \cdot x^n\), we can break down \(5^{a + 2b}\) as:
\[
5^{a + 2b} = 5^a \cdot 5^{2b}
\]

Now, let's compare this with each option:

1. **Option 1:** \(5^a \cdot 5^2 \cdot 5^b\)  
   This simplifies to \(5^{a + 2 + b}\), which is not equal to \(5^{a + 2b}\).

2. **Option 2:** \(5^a \cdot 25^b\)  
   Since \(25 = 5^2\), \(25^b\) is equivalent to \( (5^2)^b = 5^{2b}\). Therefore, \(5^a \cdot 25^b = 5^a \cdot 5^{2b} = 5^{a + 2b}\).  
   This matches \(5^{a + 2b}\), so Option 2 is correct.

3. **Option 3:** \(25^{2ab}\)  
   This does not simplify to \(5^{a + 2b}\).

4. **Option 4:** \(25^{a + 2b}\)  
   This would be equivalent to \( (5^2)^{a + 2b} = 5^{2(a + 2b)}\), which is not the same as \(5^{a + 2b}\).

The correct answer is **Option 2**: \(5^a \cdot 25^b\).

Would you like a more detailed breakdown or have any questions?

Here are some related questions to explore:
1. How do you expand powers with sums in the exponents?
2. Why does \(25^b\) become \(5^{2b}\)?
3. What other exponent properties can help simplify expressions like these?
4. How would the answer change if the base were different, like \(3^{a + 2b}\)?
5. Can we apply these rules to non-integer exponents as well?

**Tip:** When you see a composite number like \(25\) in an exponent problem, consider breaking it down into its prime base form (here, \(5^2\)) for simplification.

The expression 5^(a + 2b) is equivalent to which of the following options:
1) 5^a * 5^2 * 5^b
2) 5^a * 25^b
3) 25^(2ab)
4) 25^(a + 2b)

Learn how to find the expression equivalent to 5^(a + 2b) using properties of exponents. This problem covers exponent multiplication and simplification techniques, suitable for students in Grades 8-10.

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