If `a^(4) + a^(2)b^(2) + b^(4) = 12, a^(2) + ab+ b^(2)=4`, find ab

Answer

To solve the equations \( a^4 + a^2b^2 + b^4 = 12 \) and \( a^2 + ab + b^2 = 4 \), we will use substitution and algebraic manipulation.

### Step 1: Let \( x = ab \)

We know that \( a^2 + b^2 = (a + b)^2 - 2ab \). We can express \( a^2 + b^2 \) in terms of \( x \).

### Step 2: Rewrite \( a^2 + b^2 \)

From the second equation \( a^2 + ab + b^2 = 4 \), we can express \( a^2 + b^2 \) as:

\[ a^2 + b^2 = 4 - ab = 4 - x \]

### Step 3: Substitute into the first equation

Now, we can express \( a^4 + b^4 \) using \( a^2 + b^2 \) and \( ab \):

\[ a^4 + b^4 = (a^2 + b^2)^2 - 2(ab)^2 \]

Substituting \( a^2 + b^2 = 4 - x \):

\[ a^4 + b^4 = (4 - x)^2 - 2x^2 \]

### Step 4: Expand and simplify

Expanding \( (4 - x)^2 \):

\[ (4 - x)^2 = 16 - 8x + x^2 \]

Thus,

\[ a^4 + b^4 = 16 - 8x + x^2 - 2x^2 = 16 - 8x - x^2 \]

### Step 5: Substitute into the first equation

Now we substitute this back into the first equation:

\[ a^4 + a^2b^2 + b^4 = 12 \]

We know \( a^2b^2 = (ab)^2 = x^2 \), so:

\[ 16 - 8x - x^2 + x^2 = 12 \]

This simplifies to:

\[ 16 - 8x = 12 \]

### Step 6: Solve for \( x \)

Rearranging gives:

\[ -8x = 12 - 16 \]
\[ -8x = -4 \]
\[ x = \frac{-4}{-8} = \frac{1}{2} \]

### Conclusion

Thus, the value of \( ab \) is:

\[ \boxed{\frac{1}{2}} \]