If `a^(4) + a^(2) b^(2) +b^(4)=8, a^(2) + b^(2) + ab= 4` find ab

Answer

To solve the problem, we need to find the value of \( ab \) given the equations:

1. \( a^4 + a^2b^2 + b^4 = 8 \)
2. \( a^2 + b^2 + ab = 4 \)

We can approach this step by step.

### Step 1: Rewrite the first equation
We can express \( a^4 + b^4 + a^2b^2 \) in terms of \( a^2 + b^2 \) and \( ab \). We know that:

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

Thus, we can rewrite the first equation as:

\[ (a^2 + b^2)^2 - 2a^2b^2 + a^2b^2 = 8 \]

This simplifies to:

\[ (a^2 + b^2)^2 - a^2b^2 = 8 \]

### Step 2: Substitute \( a^2 + b^2 \)
Let \( x = a^2 + b^2 \) and \( y = ab \). From the second equation, we have:

\[ x + y = 4 \quad \text{(Equation 1)} \]

### Step 3: Substitute into the first equation
Now, substituting \( x \) and \( y \) into the modified first equation gives us:

\[ x^2 - y^2 = 8 \]

This can be factored as:

\[ (x - y)(x + y) = 8 \]

### Step 4: Substitute \( y \) from Equation 1
From Equation 1, we know \( y = 4 - x \). Substituting this into the factored equation:

\[ (x - (4 - x))(x + (4 - x)) = 8 \]

This simplifies to:

\[ (2x - 4)(4) = 8 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 4:

\[ 2x - 4 = 2 \]

Adding 4 to both sides gives:

\[ 2x = 6 \]

Dividing by 2:

\[ x = 3 \]

### Step 6: Find \( y \)
Now, substitute \( x \) back into Equation 1 to find \( y \):

\[ 3 + y = 4 \]

Thus:

\[ y = 4 - 3 = 1 \]

### Conclusion
Since \( y = ab \), we have:

\[ ab = 1 \]

### Final Answer
The value of \( ab \) is \( \boxed{1} \).

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