Divide `x^(6) - y^(6)` by the product of `x^(2) + xy + y^(2)` and `x - y`.

Answer

To solve the problem of dividing \( x^6 - y^6 \) by the product of \( (x^2 + xy + y^2) \) and \( (x - y) \), we can follow these steps:

### Step 1: Identify the expression to divide
We need to divide:
\[ \frac{x^6 - y^6}{(x^2 + xy + y^2)(x - y)} \]

### Step 2: Factor \( x^6 - y^6 \)
We can factor \( x^6 - y^6 \) using the difference of squares:
\[ x^6 - y^6 = (x^3 - y^3)(x^3 + y^3) \]
Next, we can further factor \( x^3 - y^3 \):
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]
Thus, we can express \( x^6 - y^6 \) as:
\[ x^6 - y^6 = (x - y)(x^2 + xy + y^2)(x^3 + y^3) \]

### Step 3: Substitute the factored form into the division
Now we can substitute this back into our division:
\[ \frac{(x - y)(x^2 + xy + y^2)(x^3 + y^3)}{(x^2 + xy + y^2)(x - y)} \]

### Step 4: Cancel the common terms
We can cancel the common terms \( (x - y) \) and \( (x^2 + xy + y^2) \):
\[ = x^3 + y^3 \]

### Final Answer
Thus, the result of the division is:
\[ x^3 + y^3 \]