What is the value of `{((a+b)^(2) - (a^(2) +b^(2)))/((a+b)^(2) - (a-b)^(2))}?`

Answer

To solve the expression \(\frac{(a+b)^2 - (a^2 + b^2)}{(a+b)^2 - (a-b)^2}\), we will follow these steps:

### Step 1: Expand the numerator
The numerator is \((a+b)^2 - (a^2 + b^2)\).

We know that:
\[ (a+b)^2 = a^2 + 2ab + b^2 \]

So, substituting this into the numerator gives:
\[ (a+b)^2 - (a^2 + b^2) = (a^2 + 2ab + b^2) - (a^2 + b^2) \]
Now, simplify:
\[ = a^2 + 2ab + b^2 - a^2 - b^2 = 2ab \]

### Step 2: Expand the denominator
The denominator is \((a+b)^2 - (a-b)^2\).

Using the expansion:
\[ (a-b)^2 = a^2 - 2ab + b^2 \]
Thus, we have:
\[ (a+b)^2 - (a-b)^2 = (a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) \]
Now, simplify:
\[ = a^2 + 2ab + b^2 - a^2 + 2ab - b^2 = 4ab \]

### Step 3: Substitute back into the expression
Now we can substitute our results back into the original expression:
\[ \frac{(a+b)^2 - (a^2 + b^2)}{(a+b)^2 - (a-b)^2} = \frac{2ab}{4ab} \]

### Step 4: Simplify the fraction
Now simplify the fraction:
\[ \frac{2ab}{4ab} = \frac{2}{4} = \frac{1}{2} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{1}{2} \]