The expression \( 27a^3 + 64b^3 \) can be factored using the sum of cubes formula, which is given by:

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

Here, we can identify \( x = 3a \) and \( y = 4b \) because:

\[
27a^3 = (3a)^3 \quad \text{and} \quad 64b^3 = (4b)^3
\]

So, the expression becomes:

\[
27a^3 + 64b^3 = (3a)^3 + (4b)^3
\]

Applying the sum of cubes formula:

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

Now, simplify the expression inside the second parentheses:

\[
= (3a + 4b)\left(9a^2 - 12ab + 16b^2\right)
\]

So, the fully factored form of \( 27a^3 + 64b^3 \) is:

\[
27a^3 + 64b^3 = (3a + 4b)(9a^2 - 12ab + 16b^2)
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How would you factor the difference of cubes, such as \( 27a^3 - 64b^3 \)?
2. Can you apply the sum of cubes formula to a different set of terms, like \( 8x^3 + 27y^3 \)?
3. How would you factor a sum of squares, like \( a^2 + b^2 \)?
4. What is the general formula for factoring \( a^3 + b^3 \) and \( a^3 - b^3 \)?
5. How does the sum of cubes differ from the sum of squares in terms of factorization?

**Tip:** When dealing with cubic polynomials, always check if they fit the form of a sum or difference of cubes, as this allows for straightforward factoring.

Discover how to factor the expression 27a^3 + 64b^3 using the sum of cubes formula. This solution involves key algebraic concepts and is ideal for students in Grades 9-12 looking to strengthen their understanding of polynomial factorization.

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