Find the product : `(a^(2)+b^(2))(a^(4)+b^(4))(a+b)(a-b)`.

Answer

To find the product \((a^{2}+b^{2})(a^{4}+b^{4})(a+b)(a-b)\), we can follow these steps:

### Step 1: Recognize the identities
We will use the algebraic identities:
1. \( (a+b)(a-b) = a^2 - b^2 \)
2. \( a^4 + b^4 = (a^2 + b^2)^2 - (ab)^2 \)

### Step 2: Simplify \((a+b)(a-b)\)
Using the first identity:
\[ (a+b)(a-b) = a^2 - b^2 \]

### Step 3: Substitute back into the product
Now, we can rewrite the original expression:
\[ (a^{2}+b^{2})(a^{4}+b^{4})(a+b)(a-b) = (a^{2}+b^{2})(a^{4}+b^{4})(a^2 - b^2) \]

### Step 4: Simplify \(a^{4}+b^{4}\)
Using the second identity:
\[ a^4 + b^4 = (a^2 + b^2)^2 - (ab)^2 \]

### Step 5: Substitute \(a^4 + b^4\) into the expression
Now substitute this back into our expression:
\[ = (a^{2}+b^{2})((a^{2}+b^{2})^2 - (ab)^2)(a^2 - b^2) \]

### Step 6: Let \(x = a^2 + b^2\)
Let \(x = a^2 + b^2\). Then the expression becomes:
\[ = x(x^2 - (ab)^2)(a^2 - b^2) \]

### Step 7: Expand the expression
Now we can expand:
\[ = x(a^2 - b^2)(x^2 - (ab)^2) \]

### Step 8: Final expression
The final expression can be simplified further, but the product can be expressed as:
\[ = (a^2 + b^2)(a^2 - b^2)((a^2 + b^2)^2 - (ab)^2) \]

### Conclusion
Thus, the product \((a^{2}+b^{2})(a^{4}+b^{4})(a+b)(a-b)\) simplifies to:
\[ = (a^2 + b^2)(a^2 - b^2)((a^2 + b^2)^2 - (ab)^2) \]