If ` 3 a ^(2) = b^(2) ne 0 ` then the value of ` (( a + b) ^(3) - ( a - b)^(3))/((a + b) ^(2) + (a - b)^(2))` is

Options

A. `(3b)/(2)`
B. b
C. `(b)/(2)`
D. `(2b)/(3)`

Answer

To solve the problem, we need to evaluate the expression:

\[ \frac{(a + b)^3 - (a - b)^3}{(a + b)^2 + (a - b)^2} \]

Given that \(3a^2 = b^2 \neq 0\), we can proceed with the solution step by step.

### Step 1: Expand the Numerator

We start by expanding the cubes in the numerator:

\[ (a + b)^3 = a^3 + b^3 + 3a^2b + 3ab^2 \]

\[ (a - b)^3 = a^3 - b^3 - 3a^2b + 3ab^2 \]

Now, substituting these expansions into the numerator:

\[ (a + b)^3 - (a - b)^3 = (a^3 + b^3 + 3a^2b + 3ab^2) - (a^3 - b^3 - 3a^2b + 3ab^2) \]

### Step 2: Simplify the Numerator

When we simplify the expression, we notice that \(a^3\) cancels out:

\[ = b^3 + 3a^2b + 3ab^2 + 3a^2b - b^3 - 3ab^2 \]

This simplifies to:

\[ = 6a^2b \]

### Step 3: Expand the Denominator

Next, we expand the squares in the denominator:

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

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

Now, substituting these expansions into the denominator:

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

### Step 4: Simplify the Denominator

When we simplify the expression, we notice:

\[ = 2a^2 + 2b^2 \]

### Step 5: Substitute Back into the Expression

Now, we can substitute the simplified numerator and denominator back into the original expression:

\[ \frac{6a^2b}{2a^2 + 2b^2} \]

### Step 6: Factor the Denominator

We can factor out a 2 from the denominator:

\[ = \frac{6a^2b}{2(a^2 + b^2)} = \frac{3a^2b}{a^2 + b^2} \]

### Step 7: Substitute for \(b^2\)

Since we know that \(b^2 = 3a^2\) from the given condition, we can substitute this into our expression:

\[ = \frac{3a^2b}{a^2 + 3a^2} = \frac{3a^2b}{4a^2} = \frac{3b}{4} \]

### Final Answer

Thus, the value of the expression is:

\[ \frac{3b}{4} \]