If m and n are the two natural number such that `m^(n)=25`, then find the value of `n^(m)`:-

Answer

To solve the problem, we need to find the values of \( m \) and \( n \) given that \( m^n = 25 \), and then calculate \( n^m \).

### Step-by-Step Solution:

1. **Understanding the Equation**:
We start with the equation \( m^n = 25 \). We know that \( 25 \) can be expressed as \( 5^2 \). This gives us a clue about possible values for \( m \) and \( n \).

2. **Identifying Possible Values**:
Since \( m \) and \( n \) are natural numbers, we can consider the following pairs:
- If \( m = 5 \), then \( n \) must be \( 2 \) because \( 5^2 = 25 \).
- If \( m = 25 \), then \( n \) must be \( 1 \) because \( 25^1 = 25 \).

Thus, the pairs \((m, n)\) can be \((5, 2)\) or \((25, 1)\).

3. **Calculating \( n^m \)**:
- For the pair \((m, n) = (5, 2)\):
\[ n^m = 2^5 = 32 \]
- For the pair \((m, n) = (25, 1)\):
\[ n^m = 1^{25} = 1 \]

4. **Final Result**:
The values of \( n^m \) can either be \( 32 \) or \( 1 \). Since the problem does not specify which pair to use, both answers are valid. However, the more common interpretation in such problems is to use the pair \((5, 2)\).

Thus, the final answer is:
\[ n^m = 32 \]