The given expression is:

\[
\frac{x - y}{x^3 - y^3}
\]

### Step 1: Factor the denominator
We can use the **difference of cubes** formula to factor the denominator. The difference of cubes formula is:

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

Here, \(x^3 - y^3\) is a difference of cubes, so applying the formula gives:

\[
x^3 - y^3 = (x - y)(x^2 + xy + y^2)
\]

### Step 2: Simplify the expression
Now, substitute this factorization into the original expression:

\[
\frac{x - y}{(x - y)(x^2 + xy + y^2)}
\]

Since \(x - y\) appears in both the numerator and denominator, and assuming \(x \neq y\) (to avoid division by zero), you can cancel \(x - y\) from both sides:

\[
\frac{1}{x^2 + xy + y^2}
\]

### Final result:
The simplified form of the expression is:

\[
\frac{1}{x^2 + xy + y^2}
\]

Would you like any more details or further explanation on this? 

Here are 5 related questions to expand on this:

1. What happens if \(x = y\)? 
2. How would the expression change if we had \(x^3 + y^3\) instead of \(x^3 - y^3\)?
3. Can you explain the sum of cubes formula?
4. How do you factor more complex polynomials involving cubes?
5. How does the factorization of higher degree polynomials work?

**Tip:** When factoring polynomials, always look for common patterns like the difference of squares or cubes to simplify the expressions easily.

Learn how to simplify the expression x - y divided by x³ - y³ using the difference of cubes formula. This detailed guide walks through each step, from factoring the denominator to simplifying the final expression. Suitable for students in Grades 9-11.

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