If `a^(4) + b^(4) = a^(2) b^(2)` find `a^(6) + b^(6)`

Answer

To solve the equation \( a^4 + b^4 = a^2 b^2 \) and find \( a^6 + b^6 \), we can follow these steps:

### Step 1: Rewrite the Given Equation
We start with the equation:
\[ a^4 + b^4 = a^2 b^2 \]

### Step 2: Add \( 2a^2b^2 \) to Both Sides
To manipulate the equation, we add \( 2a^2b^2 \) to both sides:
\[ a^4 + b^4 + 2a^2b^2 = a^2b^2 + 2a^2b^2 \]
This simplifies to:
\[ a^4 + b^4 + 2a^2b^2 = 3a^2b^2 \]

### Step 3: Factor the Left Side
The left side can be factored as:
\[ (a^2 + b^2)^2 = 3a^2b^2 \]

### Step 4: Take the Square Root
Taking the square root of both sides gives us:
\[ a^2 + b^2 = \sqrt{3}ab \]

### Step 5: Find \( a^6 + b^6 \)
We can use the identity for the sum of cubes:
\[ a^6 + b^6 = (a^2 + b^2)(a^4 - a^2b^2 + b^4) \]
From our earlier steps, we know:
- \( a^2 + b^2 = \sqrt{3}ab \)
- We also have \( a^4 + b^4 = a^2b^2 \), thus \( a^4 - a^2b^2 + b^4 = (a^4 + b^4) - a^2b^2 = 0 \)

### Step 6: Substitute Values
Substituting these values into the identity:
\[ a^6 + b^6 = (a^2 + b^2)(a^4 - a^2b^2 + b^4) = (\sqrt{3}ab)(0) = 0 \]

### Final Result
Thus, we conclude that:
\[ a^6 + b^6 = 0 \]