To factor the expression $x^3 - 64$ using the sum and difference of cubes, we follow these steps:

Step 1: Identify the cubes

The given expression is $x^3 - 64$.

We recognize that:

• $x^3$ is the cube of $x$ (i.e., $x^3 = (x)^3$).
• $64$ is the cube of $4$ (i.e., $64 = 4^3$).

Step 2: Apply the difference of cubes formula

The difference of cubes formula is:

$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

In this case, $a = x$ and $b = 4$. Substituting these into the formula:

$x^3 - 4^3 = (x - 4)(x^2 + 4x + 16)$

Final Answer

So, the factorization of $x^3 - 64$ is:

$x^3 - 64 = (x - 4)(x^2 + 4x + 16)$

Would you like more details or have any questions?

Here are some related questions you might find helpful:

1. How do you factor the sum of cubes $x^3 + 27$?
2. What is the general formula for factoring a sum or difference of cubes?
3. Can you explain how to recognize when an expression is a sum or difference of cubes?
4. How can you verify the factorization of $x^3 - 64$?
5. What is the next step after factoring a difference of cubes?
6. Are there any real-world applications for factoring cubes in mathematics?
7. How does factoring cubes relate to solving polynomial equations?
8. Can the expression $x^3 - 64$ be further factored?

Tip: Always check if the expression is a perfect cube before applying the sum or difference of cubes formula.