https://file.aiquickdraw.com/mathbot/file/1726931751740b7fvazhi.jpg

The expression shown in the image is:

\[
24a^3 + 81b^3
\]

This is a sum of cubes, where \( 24a^3 \) and \( 81b^3 \) are both cube terms. To factor this expression, we can use the sum of cubes formula:

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

We need to express \( 24a^3 \) and \( 81b^3 \) as perfect cubes:

- \( 24a^3 = (2a)^3 \)
- \( 81b^3 = (3b)^3 \)

So we can rewrite the expression as:

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

Now, applying the sum of cubes formula:

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

Let's simplify the terms inside the second factor:

\[
(2a)^2 = 4a^2, \quad (2a)(3b) = 6ab, \quad (3b)^2 = 9b^2
\]

Thus, the factored form is:

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

This is the complete factorization of the expression \( 24a^3 + 81b^3 \).

Would you like to ask any further details or clarifications?

### Related questions:
1. What is the general formula for factoring a sum of cubes?
2. How would you factor a difference of cubes?
3. Can you explain why the middle term in the sum of cubes formula has a negative sign?
4. What would the factored form look like for a different set of cube terms, e.g., \( 8x^3 + 27y^3 \)?
5. How can you verify that the factorization is correct?

### Tip:
When factoring sums or differences of cubes, always check if the terms are perfect cubes before applying the formula.

Learn how to factor the expression 24a^3 + 81b^3 using the sum of cubes formula. This solution breaks down the steps to express the terms as perfect cubes and apply the formula. Suitable for students in Grades 9-12.

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