If ` ( a^(2) + b^(2))^(3) = (a^(3) + b^(3))^(2)` then `(a)/(b) + (b)/(a)` is

Options

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

Answer

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

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

**Left Side:**
Using the identity \( (x + y)^3 = x^3 + y^3 + 3xy(x + y) \), we can express \( (a^2 + b^2)^3 \) as:
\[ (a^2 + b^2)^3 = a^6 + b^6 + 3a^2b^2(a^2 + b^2) \]

**Right Side:**
Using the identity \( (x + y)^2 = x^2 + y^2 + 2xy \), we can express \( (a^3 + b^3)^2 \) as:
\[ (a^3 + b^3)^2 = (a^3)^2 + (b^3)^2 + 2(a^3)(b^3) = a^6 + b^6 + 2a^3b^3 \]

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

### 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: Divide both sides by \( a^2b^2 \) (assuming \( a, b \neq 0 \))
\[ 3(a^2 + b^2) = 2ab \]

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

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

### Step 7: Multiply through by \( \frac{1}{3} \)
To isolate \( \frac{a}{b} + \frac{b}{a} \), we divide the entire equation by 3:
\[ \frac{a}{b} + \frac{b}{a} = \frac{2}{3} \]

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

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