Factor 27, a, cubed, plus, 8, b, cubed27*a*3+8*b*3 completely.

To factor the expression \( 27a^3 + 8b^3 \), we recognize that this is a sum of cubes. The formula for factoring a sum of cubes is:

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

Here, \( 27a^3 \) can be written as \( (3a)^3 \) and \( 8b^3 \) as \( (2b)^3 \). So the expression becomes:

\[
(3a)^3 + (2b)^3
\]

Applying the sum of cubes formula, we get:

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

Now simplify the second factor:

\[
(3a + 2b)\left(9a^2 - 6ab + 4b^2\right)
\]

Thus, the factored form of \( 27a^3 + 8b^3 \) is:

\[
(3a + 2b)(9a^2 - 6ab + 4b^2)
\]

Would you like more details or have any questions?

Here are some related questions:
1. How would you factor \( 64x^3 + 1 \)?
2. Can you apply the difference of cubes formula to \( 125y^3 - 27z^3 \)?
3. How do you factor \( a^3 + b^3 \) in general?
4. How does factoring cubes relate to factoring squares?
5. What happens if you try to factor \( 27a^3 - 8b^3 \)?

**Tip**: For sum and difference of cubes, memorize their factorization formulas to easily break down these expressions.

Learn how to factor the expression 27a^3 + 8b^3 using the sum of cubes formula. This problem demonstrates how to break down cubic expressions and is suitable for students in Grades 9-12.

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