If m is a positive integer, which of the following is not equal to `(2^(4))^(m)`?

Options

A. `2^(4m)`
B. `4^(2m)`
C. `2^(m)(2^(3m))`
D. `4^(m)(2^(m))`

Answer

To solve the problem, we need to determine which of the given options is not equal to \((2^4)^m\).

### Step-by-Step Solution:

1. **Simplify the expression \((2^4)^m\)**:
\[ (2^4)^m = 2^{4m} \]
This follows the power of a power rule, which states that \((a^m)^n = a^{m \cdot n}\).

**Hint**: Remember that when you raise a power to another power, you multiply the exponents.

2. **Evaluate the options**:
- **Option 1**: \(2^{4m}\)
- This is equal to \(2^{4m}\), so it is equal.

- **Option 2**: \(4^{2m}\)
- Rewrite \(4\) as \(2^2\):
\[ 4^{2m} = (2^2)^{2m} = 2^{4m} \]
- This is also equal to \(2^{4m}\).

- **Option 3**: \(2^{m} \cdot 2^{3m}\)
- Since the bases are the same, we can add the exponents:
\[ 2^{m} \cdot 2^{3m} = 2^{m + 3m} = 2^{4m} \]
- This is equal to \(2^{4m}\).

- **Option 4**: \(4^{m}\)
- Rewrite \(4\) as \(2^2\):
\[ 4^{m} = (2^2)^{m} = 2^{2m} \]
- This is not equal to \(2^{4m}\).

3. **Conclusion**:
The option that is not equal to \((2^4)^m\) is **Option 4: \(4^{m}\)**.

### Final Answer:
**Option 4: \(4^{m}\)** is not equal to \((2^4)^m\).