Identifying the Sum of Cubes Expression

Oct 13, 2025 by miodeya.com 88 views

In mathematics, a sum of cubes is a mathematical expression that can be factored into the product of three binomials. This type of expression is commonly encountered in algebra and is used to simplify complex equations. In this article, we will explore the concept of a sum of cubes and identify which of the given expressions is a sum of cubes.

What is a Sum of Cubes?

A sum of cubes is a mathematical expression of the form $a^3 + b^3$ , where $a$ and $b$ are algebraic expressions. This type of expression can be factored into the product of three binomials using the formula:

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

This formula allows us to simplify complex expressions and solve equations involving cubes.

Identifying the Sum of Cubes Expression

To identify which of the given expressions is a sum of cubes, we need to examine each option carefully. Let's start by analyzing each expression:

Option A: $-24a^{15} + 125b^{18}$

This expression can be rewritten as:

-24a^{15} + 125b^{18} = (-8a^5)^3 + (5b^6)^3

Using the formula for the sum of cubes, we can factor this expression as:

(-8a^5)^3 + (5b^6)^3 = ((-8a^5) + (5b^6))((-8a^5)^2 - (-8a^5)(5b^6) + (5b^6)^2)

This expression can be further simplified to:

(-8a^5 + 5b^6)((-8a^5)^2 + 40a^{10}b^6 + (5b^6)^2)

However, this expression does not match the form of a sum of cubes, as it contains a squared term.

Option B: $-64a^{27} + b^8$

This expression can be rewritten as:

-64a^{27} + b^8 = (-4a^9)^3 + (b^4)^2

However, this expression is not a sum of cubes, as it contains a squared term.

Option C: $27x^9 + y^6$

This expression can be rewritten as:

27x^9 + y^6 = (3x^3)^3 + (y^2)^3

Using the formula for the sum of cubes, we can factor this expression as:

(3x^3)^3 + (y^2)^3 = ((3x^3) + (y^2))((3x^3)^2 - (3x^3)(y^2) + (y^2)^2)

This expression can be further simplified to:

(3x^3 + y^2)(9x^6 - 3x^3y^2 + y^4)

This expression matches the form of a sum of cubes.

Option D: $81x^{24} + 8y^{40}$

This expression can be rewritten as:

81x^{24} + 8y^{40} = (3x^8)^3 + (2y^{20})^2

However, this expression is not a sum of cubes, as it contains a squared term.

Conclusion

In conclusion, the expression that is a sum of cubes is Option C: $27x^9 + y^6$ . This expression can be factored into the product of three binomials using the formula for the sum of cubes.

Key Takeaways

A sum of cubes is a mathematical expression that can be factored into the product of three binomials.
The formula for the sum of cubes is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ .
To identify a sum of cubes expression, look for expressions of the form $a^3 + b^3$ .
The expression $27x^9 + y^6$ is a sum of cubes and can be factored into the product of three binomials.

Q: What is the formula for the sum of cubes?

A: The formula for the sum of cubes is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ .

Q: How do I identify a sum of cubes expression?

A: To identify a sum of cubes expression, look for expressions of the form $a^3 + b^3$ . This type of expression can be factored into the product of three binomials using the formula for the sum of cubes.

Q: Can I factor a sum of cubes expression into the product of three binomials?

A: Yes, you can factor a sum of cubes expression into the product of three binomials using the formula for the sum of cubes. The formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ .

Q: What is the difference between a sum of cubes and a difference of cubes?

A: A sum of cubes is an expression of the form $a^3 + b^3$ , while a difference of cubes is an expression of the form $a^3 - b^3$ . The formula for the difference of cubes is $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ .

Q: Can I use the formula for the sum of cubes to factor a difference of cubes expression?

A: No, you cannot use the formula for the sum of cubes to factor a difference of cubes expression. The formula for the difference of cubes is $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ , which is different from the formula for the sum of cubes.

Q: How do I simplify a sum of cubes expression?

A: To simplify a sum of cubes expression, you can use the formula for the sum of cubes to factor the expression into the product of three binomials. You can then simplify each binomial separately.

Q: Can I use the formula for the sum of cubes to simplify a difference of cubes expression?

A: No, you cannot use the formula for the sum of cubes to simplify a difference of cubes expression. The formula for the difference of cubes is $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ , which is different from the formula for the sum of cubes.

Q: What are some common applications of the sum of cubes formula?

A: The sum of cubes formula has many common applications in algebra and geometry. Some examples include:

Factoring expressions of the form $a^3 + b^3$
Simplifying expressions of the form $a^3 + b^3$
Solving equations involving cubes
Proving geometric theorems involving cubes

Q: Can I use the sum of cubes formula to solve equations involving cubes?

A: Yes, you can use the sum of cubes formula to solve equations involving cubes. The formula can be used to factor and simplify expressions involving cubes, which can then be used to solve equations.

Q: What are some common mistakes to avoid when using the sum of cubes formula?

A: Some common mistakes to avoid when using the sum of cubes formula include:

Using the formula for the sum of cubes to factor a difference of cubes expression
Using the formula for the difference of cubes to factor a sum of cubes expression
Failing to simplify each binomial separately after factoring the expression
Failing to check for errors in the factoring process

Conclusion

In conclusion, the sum of cubes formula is a powerful tool for factoring and simplifying expressions involving cubes. By understanding the formula and its applications, you can solve a wide range of problems in algebra and geometry. Remember to avoid common mistakes and to use the formula carefully to ensure accurate results.

Key Takeaways

The formula for the sum of cubes is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ .
To identify a sum of cubes expression, look for expressions of the form $a^3 + b^3$ .
The formula for the sum of cubes can be used to factor and simplify expressions involving cubes.
The formula for the difference of cubes is $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ .
Common mistakes to avoid when using the sum of cubes formula include using the formula for the sum of cubes to factor a difference of cubes expression and failing to simplify each binomial separately after factoring the expression.