\( ( A + B ) ^ { - 2 } = \frac { 1 } { A ^ { 2 } } + \frac { 1 } { B ^ { 2 } } \)

Let's solve the equation step by step: Given the equation: \[ ( A + B ) ^ { - 2 } = \frac { 1 } { A ^ { 2 } } + \frac { 1 } { B ^ { 2 } } \] ### Step 1: Rewrite the left side The left side can be rewritten as: \[ \frac{1}{(A + B)^2} \] So the equation becomes: \[ \frac{1}{(A + B)^2} = \frac{1}{A^2} + \frac{1}{B^2} \] ### Step 2: Find a common denominator for the right side The common denominator for the right side is \(A^2B^2\). Thus, we can rewrite the right side as: \[ \frac{B^2 + A^2}{A^2B^2} \] ### Step 3: Set the two sides equal Now we have: \[ \frac{1}{(A + B)^2} = \frac{A^2 + B^2}{A^2B^2} \] ### Step 4: Cross-multiply Cross-multiplying gives us: \[ A^2B^2 = (A + B)^2 (A^2 + B^2) \] ### Step 5: Expand both sides Expanding the right side: \[ (A + B)^2 = A^2 + 2AB + B^2 \] So we have: \[ A^2B^2 = (A^2 + 2AB + B^2)(A^2 + B^2) \] ### Step 6: Expand the right side further Now we expand: \[ (A^2 + 2AB + B^2)(A^2 + B^2) = A^4 + A^2B^2 + 2A^3B + 2AB^3 + B^4 \] ### Step 7: Set the equation Now we have: \[ A^2B^2 = A^4 + 2A^3B + 2AB^3 + B^4 \] ### Step 8: Rearranging the equation Rearranging gives us: \[ 0 = A^4 + 2A^3B + 2AB^3 + B^4 - A^2B^2 \] This is a polynomial equation in terms of \(A\) and \(B\). ### Conclusion The equation is complex and may not yield a simple solution for \(A\) in terms of \(B\) without specific values. If you have specific values for \(B\), we can substitute and solve for \(A\). Otherwise, this is the general form of the equation derived from the original statement.