If a and b are non zero rational unequal numbers, then `((a+b)^(2)-(a-b)^(2))/(a^(2)b-ab^(2))` is equal to :

Options

A. `(1)/(a-b)`
B. `(2)/(a-b)`
C. `(4)/(a-b)`
D. `(1)/(ab)`

Answer

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

### Step 1: Expand the Numerator
First, we need to expand \((a+b)^2\) and \((a-b)^2\).

\[ (a+b)^2 = a^2 + 2ab + b^2 \]
\[ (a-b)^2 = a^2 - 2ab + b^2 \]

### Step 2: Substitute the Expansions into the Numerator
Now, substitute these expansions into the numerator:

\[ (a+b)^2 - (a-b)^2 = (a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) \]

### Step 3: Simplify the Numerator
Now simplify the expression:

\[ = a^2 + 2ab + b^2 - a^2 + 2ab - b^2 \]
\[ = 4ab \]

### Step 4: Expand the Denominator
Next, we need to simplify the denominator \(a^2b - ab^2\):

\[ a^2b - ab^2 = ab(a - b) \]

### Step 5: Combine the Results
Now we can substitute the simplified numerator and denominator back into the original expression:

\[ \frac{(a+b)^2 - (a-b)^2}{a^2b - ab^2} = \frac{4ab}{ab(a-b)} \]

### Step 6: Simplify the Expression
Now, we can simplify the fraction:

\[ = \frac{4ab}{ab(a-b)} = \frac{4}{a-b} \]

### Final Result
Thus, the expression simplifies to:

\[ \frac{4}{a-b} \]