If `(a ^(2) + b ^(2)) ^(3) = (a ^(3) + b ^(3)) ^(2) and ab ne 0,` then `((a)/(b) + (b)/(a)) ^(6)` is equal to

Options

A. `(a ^(6) + b ^(6))/( a ^(3) b ^(3))`
B. `(64)/(729)`
C. 1
D. `(a ^(6) + a ^(3) b ^(3) + b ^(6))/( a ^(2) b ^(4) + a ^(4) b ^(2))`

Answer

To solve the equation \((a^2 + b^2)^3 = (a^3 + b^3)^2\) under the condition that \(ab \neq 0\), we need to manipulate the expressions step by step.

### Step 1: Expand both sides
We start by expanding both sides of the equation using known algebraic identities.

1. **Left Side**:
\[ (a^2 + b^2)^3 = a^6 + b^6 + 3a^2b^2(a^2 + b^2) \]

2. **Right Side**:
\[ (a^3 + b^3)^2 = (a^3 + b^3)(a^3 + b^3) = a^6 + b^6 + 2a^3b^3 \]

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

### Step 3: Cancel common terms
We can cancel \(a^6 + b^6\) from both sides:
\[ 3a^2b^2(a^2 + b^2) = 2a^3b^3 \]

### Step 4: Rearrange the equation
Now, we can rearrange the equation:
\[ 3a^2b^2(a^2 + b^2) - 2a^3b^3 = 0 \]

### Step 5: Factor the equation
Factoring out common terms:
\[ a^2b^2(3(a^2 + b^2) - 2ab) = 0 \]

Since \(ab \neq 0\), we can ignore the factor \(a^2b^2\), leading us to:
\[ 3(a^2 + b^2) - 2ab = 0 \]

### Step 6: Solve for \(a^2 + b^2\)
Rearranging gives:
\[ 3(a^2 + b^2) = 2ab \]
\[ a^2 + b^2 = \frac{2}{3}ab \]

### Step 7: Find \(\left(\frac{a}{b} + \frac{b}{a}\right)^6\)
Now we need to find \(\left(\frac{a}{b} + \frac{b}{a}\right)^6\):
\[ \frac{a}{b} + \frac{b}{a} = \frac{a^2 + b^2}{ab} \]
Substituting \(a^2 + b^2 = \frac{2}{3}ab\):
\[ \frac{a^2 + b^2}{ab} = \frac{\frac{2}{3}ab}{ab} = \frac{2}{3} \]

### Step 8: Raise to the power of 6
Now we raise this to the power of 6:
\[ \left(\frac{2}{3}\right)^6 = \frac{2^6}{3^6} = \frac{64}{729} \]

### Final Answer
Thus, the value of \(\left(\frac{a}{b} + \frac{b}{a}\right)^6\) is:
\[ \frac{64}{729} \]