If `m+ (1)/(m+2)= 0` find
`(m+2)^(112) + (1)/((m+2)^(112))=` ?

Answer

To solve the equation \( m + \frac{1}{m + 2} = 0 \) and find the value of \( (m + 2)^{112} + \frac{1}{(m + 2)^{112}} \), we can follow these steps:

### Step 1: Solve for \( m + 2 \)

Starting with the given equation:

\[ m + \frac{1}{m + 2} = 0 \]

We can rearrange this equation by isolating the fraction:

\[ \frac{1}{m + 2} = -m \]

### Step 2: Multiply both sides by \( m + 2 \)

To eliminate the fraction, we multiply both sides by \( m + 2 \):

\[ 1 = -m(m + 2) \]

This simplifies to:

\[ 1 = -m^2 - 2m \]

### Step 3: Rearrange the equation

Rearranging gives us:

\[ m^2 + 2m + 1 = 0 \]

### Step 4: Factor the quadratic equation

This can be factored as:

\[ (m + 1)^2 = 0 \]

### Step 5: Solve for \( m \)

Setting the factor equal to zero gives:

\[ m + 1 = 0 \implies m = -1 \]

### Step 6: Find \( m + 2 \)

Now, substituting \( m \) back into \( m + 2 \):

\[ m + 2 = -1 + 2 = 1 \]

### Step 7: Substitute into the expression

Now we substitute \( m + 2 \) into the expression we need to evaluate:

\[ (m + 2)^{112} + \frac{1}{(m + 2)^{112}} = 1^{112} + \frac{1}{1^{112}} \]

### Step 8: Simplify the expression

This simplifies to:

\[ 1 + 1 = 2 \]

### Final Answer

Thus, the value of \( (m + 2)^{112} + \frac{1}{(m + 2)^{112}} \) is:

\[ \boxed{2} \]

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