To factorize the expression \( 27x^3 - 64 \), recognize that this is a **difference of cubes** since \( 27x^3 = (3x)^3 \) and \( 64 = 4^3 \). The general formula for factoring a difference of cubes is:

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

Here, \( a = 3x \) and \( b = 4 \). Applying the formula:

\[
27x^3 - 64 = (3x - 4)((3x)^2 + (3x)(4) + 4^2)
\]

Now, simplify each term:

\[
= (3x - 4)(9x^2 + 12x + 16)
\]

Thus, the factorization of \( 27x^3 - 64 \) is:

\[
(3x - 4)(9x^2 + 12x + 16)
\]

Would you like further details or have any questions?

Here are some related questions:
1. How do you factor the sum of cubes?
2. What is the importance of recognizing perfect cubes in algebraic expressions?
3. How can you solve equations involving cubic expressions?
4. Can you graph cubic functions and explain their key features?
5. What are some real-world applications of cubic equations?

**Tip:** Always check for special factorization patterns like the difference or sum of cubes to simplify complex expressions efficiently.

Learn how to factorize the expression 27x^3 - 64 using the difference of cubes formula. This problem involves algebraic manipulation of cubic equations and introduces the difference of cubes factorization method. Suitable for advanced high school students and college algebra courses.

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