To factor the expression \(125a^3 - 64b^3\), we can recognize that it is a difference of cubes. The formula for factoring a difference of cubes is:

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

Here, we have:
- \(x = 5a\) (since \((5a)^3 = 125a^3\))
- \(y = 4b\) (since \((4b)^3 = 64b^3\))

Applying the formula:

\[
125a^3 - 64b^3 = (5a - 4b)((5a)^2 + (5a)(4b) + (4b)^2)
\]

Now, let's simplify each term:

1. \((5a)^2 = 25a^2\)
2. \((5a)(4b) = 20ab\)
3. \((4b)^2 = 16b^2\)

So, the factored form is:

\[
125a^3 - 64b^3 = (5a - 4b)(25a^2 + 20ab + 16b^2)
\]

Would you like me to explain any part of the process in more detail?

Here are 5 related questions:
1. How would you factor a sum of cubes, such as \(x^3 + y^3\)?
2. Can you simplify the expression \((5a - 4b)^2\)?
3. What is the general formula for factoring cubes?
4. How can the difference of cubes formula be derived?
5. How would you solve the equation \(125a^3 - 64b^3 = 0\) for \(a\) and \(b\)?

**Tip:** Remember that recognizing common patterns like the difference of squares and cubes can make factoring much easier.

Learn how to factor the expression 125a^3 − 64b^3 using the difference of cubes formula. This problem demonstrates key algebraic techniques for factoring polynomials, suitable for students in Grades 9-12.

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