Factorize `x^9-y^9`

Answer

To factorize the expression \( x^9 - y^9 \), we can follow these steps:

### Step 1: Recognize the Difference of Cubes
We can rewrite \( x^9 - y^9 \) as \( (x^3)^3 - (y^3)^3 \). This allows us to use the formula for the difference of cubes, which states that:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Here, let \( a = x^3 \) and \( b = y^3 \).

### Step 2: Apply the Difference of Cubes Formula
Using the difference of cubes formula:
\[ x^9 - y^9 = (x^3 - y^3)((x^3)^2 + x^3y^3 + (y^3)^2) \]
This simplifies to:
\[ x^9 - y^9 = (x^3 - y^3)(x^6 + x^3y^3 + y^6) \]

### Step 3: Factor \( x^3 - y^3 \) Again
Now, we can further factor \( x^3 - y^3 \) using the same difference of cubes formula:
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]

### Step 4: Combine the Factors
Now, substituting back, we have:
\[ x^9 - y^9 = (x - y)(x^2 + xy + y^2)(x^6 + x^3y^3 + y^6) \]

### Final Result
Thus, the complete factorization of \( x^9 - y^9 \) is:
\[ x^9 - y^9 = (x - y)(x^2 + xy + y^2)(x^6 + x^3y^3 + y^6) \]