Find the value of k, if
`(k)/(x^(2)-4) = (1)/(x-2)-(1)/(x+2)`

Answer

To find the value of \( k \) in the equation

\[ \frac{k}{x^2 - 4} = \frac{1}{x - 2} - \frac{1}{x + 2}, \]

we can follow these steps:

### Step 1: Rewrite the left-hand side
The denominator \( x^2 - 4 \) can be factored as \( (x - 2)(x + 2) \). Therefore, we can rewrite the left-hand side as:

\[ \frac{k}{(x - 2)(x + 2)}. \]

### Step 2: Combine the right-hand side
To combine the fractions on the right-hand side, we need a common denominator, which is also \( (x - 2)(x + 2) \). Thus, we rewrite the right-hand side:

\[ \frac{1}{x - 2} - \frac{1}{x + 2} = \frac{(x + 2) - (x - 2)}{(x - 2)(x + 2)}. \]

### Step 3: Simplify the numerator
Now, simplify the numerator:

\[ (x + 2) - (x - 2) = x + 2 - x + 2 = 4. \]

So, we can rewrite the right-hand side as:

\[ \frac{4}{(x - 2)(x + 2)}. \]

### Step 4: Set the left-hand side equal to the right-hand side
Now we have:

\[ \frac{k}{(x - 2)(x + 2)} = \frac{4}{(x - 2)(x + 2)}. \]

### Step 5: Equate the numerators
Since the denominators are the same, we can equate the numerators:

\[ k = 4. \]

### Conclusion
Thus, the value of \( k \) is

\[ \boxed{4}. \]

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