If `m - (1)/(m ) = 5`, find : (i) `m^(2) + (1)/( m^2)`

Answer

To solve the equation \( m - \frac{1}{m} = 5 \) and find \( m^2 + \frac{1}{m^2} \), we can follow these steps:

### Step 1: Square both sides of the equation
Start with the equation:
\[ m - \frac{1}{m} = 5 \]
Square both sides:
\[ \left(m - \frac{1}{m}\right)^2 = 5^2 \]
This gives:
\[ m^2 - 2 \cdot m \cdot \frac{1}{m} + \frac{1}{m^2} = 25 \]

### Step 2: Simplify the equation
The middle term simplifies:
\[ m^2 - 2 + \frac{1}{m^2} = 25 \]
Now, we can rearrange this to isolate \( m^2 + \frac{1}{m^2} \):
\[ m^2 + \frac{1}{m^2} = 25 + 2 \]

### Step 3: Calculate the final result
Now, add the numbers:
\[ m^2 + \frac{1}{m^2} = 27 \]

### Final Answer
Thus, the value of \( m^2 + \frac{1}{m^2} \) is:
\[ \boxed{27} \]