`a^12-b^12`

Answer

To factor the expression \( a^{12} - b^{12} \), we can follow these steps:

### Step 1: Recognize the Difference of Squares
The expression \( a^{12} - b^{12} \) can be recognized as a difference of squares. We can use the identity:
\[ x^2 - y^2 = (x - y)(x + y) \]
In our case, we can rewrite \( a^{12} \) and \( b^{12} \) as:
\[ (a^6)^2 - (b^6)^2 \]
Thus, we can apply the difference of squares:
\[ a^{12} - b^{12} = (a^6 - b^6)(a^6 + b^6) \]

### Step 2: Factor \( a^6 - b^6 \)
Next, we need to factor \( a^6 - b^6 \) again using the difference of squares:
\[ a^6 - b^6 = (a^3)^2 - (b^3)^2 = (a^3 - b^3)(a^3 + b^3) \]
So now we have:
\[ a^{12} - b^{12} = (a^3 - b^3)(a^3 + b^3)(a^6 + b^6) \]

### Step 3: Factor \( a^3 - b^3 \) and \( a^3 + b^3 \)
Now we can use the identities for the difference and sum of cubes:
- The difference of cubes:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
- The sum of cubes:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

### Final Expression
Putting it all together, we have:
\[ a^{12} - b^{12} = (a - b)(a^2 + ab + b^2)(a + b)(a^2 - ab + b^2)(a^6 + b^6) \]

### Summary of the Factorization
Thus, the complete factorization of \( a^{12} - b^{12} \) is:
\[ a^{12} - b^{12} = (a - b)(a + b)(a^2 + ab + b^2)(a^2 - ab + b^2)(a^6 + b^6) \]