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 \((a^2 + b^2)^3 = (a^3 + b^3)^2\) and find the numerical value of \(\frac{a}{b} + \frac{b}{a}\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ (a^2 + b^2)^3 = (a^3 + b^3)^2 \]

### Step 2: Use the identity for \(a^3 + b^3\)
Recall the identity:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Thus, we can express the right-hand side:
\[ (a^3 + b^3)^2 = [(a + b)(a^2 - ab + b^2)]^2 \]

### Step 3: Expand both sides
Now, we will expand both sides:
- Left-hand side:
\[ (a^2 + b^2)^3 \]
- Right-hand side:
\[ [(a + b)(a^2 - ab + b^2)]^2 = (a + b)^2 (a^2 - ab + b^2)^2 \]

### Step 4: Set up a relationship
Since both sides are equal, we can analyze the structure of the equation. We can also use the fact that:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \implies (a^2 + b^2)^3 = [(a + b)(a^2 - ab + b^2)]^2 \]

### Step 5: Simplify using \(x = \frac{a}{b}\)
Let \(x = \frac{a}{b}\). Then, \(\frac{b}{a} = \frac{1}{x}\). We need to find:
\[ x + \frac{1}{x} \]

### Step 6: Substitute and simplify
Using the relationship we derived, we can express \(a^2 + b^2\) and \(ab\) in terms of \(x\):
\[ a^2 + b^2 = b^2(x^2 + 1) \quad \text{and} \quad ab = b^2x \]
Substituting these into the equation gives:
\[ (b^2(x^2 + 1))^3 = (b^3(x^3 + 1))^2 \]

### Step 7: Cancel \(b^6\)
Since \(b \neq 0\), we can divide both sides by \(b^6\):
\[ (x^2 + 1)^3 = (x^3 + 1)^2 \]

### Step 8: Solve for \(x\)
Now we can solve the equation:
1. Expand both sides.
2. Set them equal and simplify.

After simplification, we find:
\[ x + \frac{1}{x} = 2 \quad \text{or} \quad x + \frac{1}{x} = \frac{2}{3} \]

### Final Step: Conclusion
Thus, the numerical value of \(\frac{a}{b} + \frac{b}{a}\) is:
\[ \frac{2}{3} \]