https://file.aiquickdraw.com/mathbot/file/1729500232588411ncko2.jpg

The question is asking which expression represents a sum of cubes. To determine this, we need to identify whether the terms in each expression can be rewritten as cubes of binomials or monomials.

### Sum of cubes form:
The sum of cubes follows this pattern:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
Each term must be a perfect cube.

Let's analyze each option:

1. **Option 1:**
   \[
   -27a^3b^6 + 8a^9b^{12}
   \]
   - \(-27a^3b^6 = (-3ab^2)^3\)
   - \(8a^9b^{12} = (2a^3b^4)^3\)
   
   Both terms are cubes, but this is not a sum (due to the negative sign).

2. **Option 2:**
   \[
   -9a^3b^6 + a^9b^{10}
   \]
   - \(-9a^3b^6\) and \(a^9b^{10}\) are not cubes of any expression.

3. **Option 3:**
   \[
   9a^3b^6 + 8a^9b^{12}
   \]
   - \(9a^3b^6\) is not a perfect cube.

4. **Option 4:**
   \[
   27a^3b^6 + 8a^9b^{12}
   \]
   - \(27a^3b^6 = (3ab^2)^3\)
   - \(8a^9b^{12} = (2a^3b^4)^3\)

   Both terms are perfect cubes, and this is a sum. Therefore, **Option 4** is the correct answer.

### Conclusion:
The correct expression is **Option 4: \(27a^3b^6 + 8a^9b^{12}\)**.

Would you like more details, or do you have any questions about this? Here are five related questions to consider:

1. How can you identify if a term is a perfect cube?
2. What is the difference between a sum of cubes and a difference of cubes?
3. Can you factor the sum of cubes into binomials and trinomials?
4. Why is Option 1 incorrect, despite both terms being cubes?
5. What are some real-world applications of the sum of cubes?

**Tip:** When checking for cubes, always look for variables and coefficients raised to powers that are multiples of 3.

Learn how to identify an algebraic expression that represents a sum of cubes. This problem focuses on the formula for sum of cubes and its application in polynomial expressions. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation