The expression \( 64a^3 + 125b^3 \) can be factored using the sum of cubes formula. The sum of cubes formula is:

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

In this case, we need to identify the cubes in the given expression. We have:

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

Thus, we can rewrite the expression as:

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

Now applying the sum of cubes formula with \( a = 4a \) and \( b = 5b \):

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

Simplifying inside the parentheses:

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

So, the factored form of \( 64a^3 + 125b^3 \) is:

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

Would you like more details or have any questions?

Here are 5 numbered questions you might ask:

1. How do you factor the difference of cubes?
2. Can you provide an example using a different sum of cubes?
3. What is the sum of cubes formula used for?
4. How can you verify that the factorization is correct?
5. Can you explain the difference between the sum of cubes and the difference of cubes formulas?

**Tip:** Always look for patterns in polynomials that match standard formulas, like the sum or difference of cubes, to simplify the factorization process.

Learn how to factorize the expression 64a³ + 125b³ using the sum of cubes formula. This problem involves algebraic manipulations of polynomials, specifically focusing on the sum of cubes formula. Suitable for high school students in Grades 9-12.

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