729r3 - 64s3 = ?

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(9r+4s)(81r2-36rs+16s2)
(9r-4s)(81r2-36rs+16s2)
(9r-4s)(81r2+36rs+16s2)
(9r+4s)(81r2+36rs-16s2)

The expression given is:

\[ 729r^3 - 64s^3 \]

This is a difference of cubes, which can be factored using the formula:

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

In this case:

- \( a = 9r \) because \( (9r)^3 = 729r^3 \)
- \( b = 4s \) because \( (4s)^3 = 64s^3 \)

Applying the difference of cubes formula:

\[
729r^3 - 64s^3 = (9r - 4s)\left((9r)^2 + (9r)(4s) + (4s)^2\right)
\]

Simplifying the terms inside the parentheses:

\[
= (9r - 4s)(81r^2 + 36rs + 16s^2)
\]

The correct answer is:

**(9r - 4s)(81r\(^2\) + 36rs + 16s\(^2\))**

Do you want more details or have any questions?

Here are 8 questions you might find helpful:

1. How can the sum of cubes be factored?
2. What is the general formula for the difference of squares?
3. Can the expression \( 729r^3 + 64s^3 \) be factored?
4. How can you recognize a cubic expression that can be factored?
5. What are some applications of factoring cubic expressions?
6. How does the distributive property apply in factoring expressions?
7. What other types of polynomial expressions can be factored similarly?
8. How does factoring help in solving polynomial equations?

**Tip:** Recognizing patterns like the difference of cubes can simplify solving and factoring complex polynomial expressions.

Learn how to factor the expression 729r^3 - 64s^3 using the difference of cubes formula. This detailed solution covers algebraic concepts such as polynomials, factoring, and specific techniques like the difference of cubes theorem. Suitable for advanced high school students.

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