Factorise :
`a^(6) - b^(6)`

Answer

To factorize \( a^6 - b^6 \), we can follow these steps:

### Step 1: Recognize the difference of squares
The expression \( a^6 - b^6 \) can be recognized as a difference of squares:
\[ a^6 - b^6 = (a^3)^2 - (b^3)^2 \]

### Step 2: Apply the difference of squares formula
Using the difference of squares formula \( x^2 - y^2 = (x - y)(x + y) \), we can factor the expression:
\[ a^6 - b^6 = (a^3 - b^3)(a^3 + b^3) \]

### Step 3: Factor \( a^3 - b^3 \) and \( a^3 + b^3 \)
Now, we need to factor \( a^3 - b^3 \) and \( a^3 + b^3 \) using their respective formulas:
- The formula for \( a^3 - b^3 \) is:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
- The formula for \( a^3 + b^3 \) is:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]

### Step 4: Combine all factors
Substituting these back into our expression, we have:
\[ a^6 - b^6 = (a^3 - b^3)(a^3 + b^3) = (a - b)(a^2 + ab + b^2)(a + b)(a^2 - ab + b^2) \]

### Final Factorized Form
Thus, the complete factorization of \( a^6 - b^6 \) is:
\[ a^6 - b^6 = (a - b)(a + b)(a^2 + ab + b^2)(a^2 - ab + b^2) \]

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