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We are tasked with determining whether the expression \( 9^{0.5m} \cdot 9^{2m} \) is equivalent to each of the provided expressions:

### Step 1: Simplify \( 9^{0.5m} \cdot 9^{2m} \)

Using the rule of exponents \( a^x \cdot a^y = a^{x+y} \), we can simplify this:

\[
9^{0.5m} \cdot 9^{2m} = 9^{0.5m + 2m} = 9^{2.5m}
\]

Thus, \( 9^{0.5m} \cdot 9^{2m} = 9^{2.5m} \).

### Step 2: Analyze each of the given expressions

1. **\( 3^{5m} \)**:
   - Notice that \( 9 = 3^2 \), so we can express \( 9^{2.5m} \) as:
   \[
   9^{2.5m} = (3^2)^{2.5m} = 3^{5m}
   \]
   Therefore, \( 9^{2.5m} \) is equivalent to \( 3^{5m} \). This is **equivalent**.

2. **\( 81^m \)**:
   - We know that \( 81 = 3^4 \), so we can rewrite \( 81^m \) as:
   \[
   81^m = (3^4)^m = 3^{4m}
   \]
   Since \( 9^{2.5m} = 3^{5m} \), and \( 3^{4m} \) is not equal to \( 3^{5m} \), this is **not equivalent**.

3. **\( 9^{2.5m} \)**:
   - This expression is exactly the same as \( 9^{2.5m} \). Hence, it is **equivalent**.

### Summary of equivalence:
- \( 3^{5m} \) is **equivalent**.
- \( 81^m \) is **not equivalent**.
- \( 9^{2.5m} \) is **equivalent**.

Would you like more details or have any questions?  
Here are 5 related questions for further exploration:
1. How do exponent rules help in simplifying complex expressions?
2. Can the expression \( a^x \cdot a^y = a^{x+y} \) be applied to all types of bases?
3. How do you convert a higher power base (like \( 9 \)) to a smaller base (like \( 3 \))?
4. What is the significance of fractional exponents in these problems?
5. How can we generalize the exponent rule for different variable exponents?

**Tip:** Always simplify exponential expressions by converting bases when possible to make comparisons easier.

Determine whether 9^{0.5m} . 9^{2m} is equivalent to each of the following expressions: 3^{5m}, 81^{m}, 9^{2.5m}.

Learn how to determine if the expression 9^{0.5m} * 9^{2m} is equivalent to 3^{5m}, 81^{m}, or 9^{2.5m}. This problem covers exponent rules, simplification of exponential expressions, and base conversions. Suitable for students in Grades 9-11.

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