How do you factor x⁸ - y⁸?

To factor the expression \(x^8 - y^8\), we can recognize it as a difference of squares. The difference of squares formula states that \(a^2 - b^2 = (a - b)(a + b)\). Here, we can rewrite \(x^8 - y^8\) as follows:

Step 1: Identify the Squares

Notice that \(x^8\) can be expressed as \((x^4)^2\) and \(y^8\) as \((y^4)^2\). Thus, we can rewrite the expression:

x⁸ - y⁸ = (x⁴)² - (y⁴)²

Step 2: Apply the Difference of Squares

Now, using the difference of squares formula:

(x⁴ - y⁴)(x⁴ + y⁴)

Step 3: Factor Further

The term \(x^4 - y^4\) can also be factored further since it is again a difference of squares:

x⁴ - y⁴ = (x² - y²)(x² + y²)

Final Factored Form

Putting it all together, we have:

• x⁸ - y⁸ = (x² - y²)(x² + y²)(x⁴ + y⁴)

This is the complete factorization of \(x^8 - y^8\).