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

Options

A. `(3)/(2)`
B. `(2)/(3)`
C. 1
D. `(4)/(9)`

Answer

To solve the equation \((a^2 + b^2)^3 = (a^3 + b^3)^2\) under the condition that \(ab \neq 0\), we will follow these steps:

### Step 1: Expand both sides using identities
We will use the identities for the expansion of cubes and squares:
- \((x+y)^2 = x^2 + y^2 + 2xy\)
- \((x+y)^3 = x^3 + y^3 + 3xy(x+y)\)

For the left-hand side:
\[ (a^2 + b^2)^3 = (a^2)^3 + (b^2)^3 + 3(a^2)(b^2)(a^2 + b^2) = a^6 + b^6 + 3a^2b^2(a^2 + b^2) \]

For the right-hand side:
\[ (a^3 + b^3)^2 = (a^3)^2 + (b^3)^2 + 2(a^3)(b^3) = a^6 + b^6 + 2a^3b^3 \]

### Step 2: Set the expanded forms equal to each other
Now we set the two expanded forms equal:
\[ a^6 + b^6 + 3a^2b^2(a^2 + b^2) = a^6 + b^6 + 2a^3b^3 \]

### Step 3: Simplify the equation
Subtract \(a^6 + b^6\) from both sides:
\[ 3a^2b^2(a^2 + b^2) = 2a^3b^3 \]

### Step 4: Factor out common terms
Since \(ab \neq 0\), we can divide both sides by \(a^2b^2\):
\[ 3(a^2 + b^2) = 2ab \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ 3a^2 + 3b^2 = 2ab \]

### Step 6: Divide by \(ab\)
Now divide the entire equation by \(ab\):
\[ \frac{3a^2}{ab} + \frac{3b^2}{ab} = 2 \]
This simplifies to:
\[ 3\left(\frac{a}{b} + \frac{b}{a}\right) = 2 \]

### Step 7: Solve for \(\frac{a}{b} + \frac{b}{a}\)
Now divide both sides by 3:
\[ \frac{a}{b} + \frac{b}{a} = \frac{2}{3} \]

Thus, the numerical value of \(\frac{a}{b} + \frac{b}{a}\) is \(\frac{2}{3}\).

### Final Answer:
\[ \frac{a}{b} + \frac{b}{a} = \frac{2}{3} \]

---