If `m+ (1)/(m-2)=4`, find
`(m-2)^(111) + (1)/((m-2)^(111))=`?

Answer

To solve the equation \( m + \frac{1}{m-2} = 4 \) and find the value of \( (m-2)^{111} + \frac{1}{(m-2)^{111}} \), we can follow these steps:

### Step 1: Rearranging the equation
We start with the equation:
\[ m + \frac{1}{m-2} = 4 \]
Subtract \( m \) from both sides:
\[ \frac{1}{m-2} = 4 - m \]

### Step 2: Cross-multiplying
Next, we can cross-multiply to eliminate the fraction:
\[ 1 = (4 - m)(m - 2) \]

### Step 3: Expanding the equation
Now, expand the right-hand side:
\[ 1 = 4m - 8 - m^2 + 2m \]
Combine like terms:
\[ 1 = -m^2 + 6m - 8 \]

### Step 4: Rearranging into standard form
Rearranging gives us:
\[ m^2 - 6m + 9 = 0 \]

### Step 5: Factoring the quadratic
This can be factored as:
\[ (m - 3)^2 = 0 \]
Thus, we find:
\[ m - 3 = 0 \quad \Rightarrow \quad m = 3 \]

### Step 6: Finding \( m - 2 \)
Now we can find \( m - 2 \):
\[ m - 2 = 3 - 2 = 1 \]

### Step 7: Calculating \( (m-2)^{111} + \frac{1}{(m-2)^{111}} \)
Substituting \( m - 2 = 1 \) into the expression we need to evaluate:
\[ (m-2)^{111} + \frac{1}{(m-2)^{111}} = 1^{111} + \frac{1}{1^{111}} = 1 + 1 = 2 \]

### Final Answer
Thus, the final result is:
\[ \boxed{2} \]