If `m+ (1)/(m-2)=0`, find
`(m-2)^(12) + (1)/((m-2)^(11))=` ?

Answer

To solve the equation \( m + \frac{1}{m-2} = 0 \) and find the expression \( (m-2)^{12} + \frac{1}{(m-2)^{11}} \), we can follow these steps:

### Step 1: Rearrange the given equation
Starting with the equation:
\[ m + \frac{1}{m-2} = 0 \]
We can rearrange it to isolate \( m \):
\[ m = -\frac{1}{m-2} \]

### Step 2: Multiply both sides by \( m-2 \)
To eliminate the fraction, multiply both sides by \( m-2 \):
\[ m(m-2) = -1 \]
This expands to:
\[ m^2 - 2m + 1 = 0 \]

### Step 3: Solve the quadratic equation
Now we can solve the quadratic equation \( m^2 - 2m + 1 = 0 \). This can be factored as:
\[ (m-1)^2 = 0 \]
Thus, we find:
\[ m - 1 = 0 \implies m = 1 \]

### Step 4: Find \( m-2 \)
Now, substitute \( m = 1 \) back to find \( m-2 \):
\[ m - 2 = 1 - 2 = -1 \]

### Step 5: Substitute \( m-2 \) into the expression
Now we need to evaluate:
\[ (m-2)^{12} + \frac{1}{(m-2)^{11}} \]
Substituting \( m-2 = -1 \):
\[ (-1)^{12} + \frac{1}{(-1)^{11}} \]

### Step 6: Calculate the powers
Calculating the powers:
\[ (-1)^{12} = 1 \quad \text{(because 12 is even)} \]
\[ (-1)^{11} = -1 \quad \text{(because 11 is odd)} \]
Thus:
\[ \frac{1}{(-1)^{11}} = \frac{1}{-1} = -1 \]

### Step 7: Combine the results
Now combine the results:
\[ 1 + (-1) = 1 - 1 = 0 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{0} \]