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The expression to simplify is:

\[
3^a \times 4^0 \times 5^{2a}
\]

### Step-by-step simplification:
1. **Simplify \( 4^0 \):** 
   Any number raised to the power of 0 is 1, so:
   \[
   4^0 = 1
   \]
   Therefore, the expression becomes:
   \[
   3^a \times 1 \times 5^{2a} = 3^a \times 5^{2a}
   \]

2. **Combine the terms with the same base or apply index rules:**
   Since there are no common bases between \( 3^a \) and \( 5^{2a} \), you cannot combine them into a single base. However, you can express the product as a single expression:
   \[
   3^a \times 5^{2a} = (3 \times 5^2)^a = 3^a \times 25^a
   \]

   So the simplified expression becomes:
   \[
   (3 \times 25)^a = 75^a
   \]

Thus, the expression simplifies to:

\[
\boxed{75^a}
\]

Would you like further details on any steps?

### Follow-up questions:
1. Why is \( 4^0 \) equal to 1?
2. How do you combine powers with the same base?
3. Why can't we combine \( 3^a \) and \( 5^{2a} \) directly?
4. What is the general rule for multiplying powers with different bases?
5. Can you provide another example of simplifying similar exponential expressions?

### Tip:
When simplifying expressions with exponents, always check if any term can be reduced first, such as powers of 0 or 1.

Express 3^a * 4^0 * 5^2a as a single index.

Learn how to simplify the expression 3^a * 4^0 * 5^2a by applying exponent rules and expressing it as a single exponent. This problem involves key concepts like exponentiation rules and powers of zero. Suitable for students in Grades 8-10.

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