What Is $64x^6 + 27$ Written As A Sum Of Cubes?A. $(4x)^3 + 3^3$B. $\left(4x^2\right)^3 + 3^3$C. \$\left(4x^2\right)^3 + 9^3$[/tex\]

Introduction to Sum of Cubes

A sum of cubes is a mathematical expression that can be factored into the product of three binomials. It is a fundamental concept in algebra and is used to simplify complex expressions. The general form of a sum of cubes is $a^3 + b^3$, where $a$ and $b$ are any real numbers. In this article, we will explore how to write the expression $64x^6 + 27$ as a sum of cubes.

Understanding the Expression

The given expression is $64x^6 + 27$. To write this expression as a sum of cubes, we need to identify the values of $a$ and $b$ in the general form $a^3 + b^3$. We can see that $64x^6$ is a perfect cube, as it can be written as $(4x^2)^3$. Similarly, $27$ is also a perfect cube, as it can be written as $3^3$.

Writing the Expression as a Sum of Cubes

Using the fact that $64x^6$ is $(4x^2)^3$ and $27$ is $3^3$, we can write the expression $64x^6 + 27$ as a sum of cubes:

$$64x^6 + 27 = (4x^2)^3 + 3^3$$

This is the correct form of a sum of cubes, where $a = 4x^2$ and $b = 3$.

Comparing with the Options

Now, let's compare our result with the given options:

A. $(4x)^3 + 3^3$ B. $\left(4x^2\right)3 + 3^3$ C. $\left(4x^2\right)3 + 9^3$

We can see that option B matches our result exactly. Therefore, the correct answer is:

B. $\left(4x^2\right)3 + 3^3$

Conclusion

In this article, we have explored how to write the expression $64x^6 + 27$ as a sum of cubes. We have identified the values of $a$ and $b$ in the general form $a^3 + b^3$ and used this information to write the expression as a sum of cubes. Our result matches option B, which is the correct answer.

Frequently Asked Questions

• What is a sum of cubes? A sum of cubes is a mathematical expression that can be factored into the product of three binomials. It is a fundamental concept in algebra and is used to simplify complex expressions.
• How do I write an expression as a sum of cubes? To write an expression as a sum of cubes, you need to identify the values of $a$ and $b$ in the general form $a^3 + b^3$. You can then use this information to factor the expression into the product of three binomials.
• What is the general form of a sum of cubes? The general form of a sum of cubes is $a^3 + b^3$, where $a$ and $b$ are any real numbers.

Final Answer

The final answer is B. $\left(4x^2\right)3 + 3^3$.

Introduction

In our previous article, we explored how to write the expression $64x^6 + 27$ as a sum of cubes. We identified the values of $a$ and $b$ in the general form $a^3 + b^3$ and used this information to write the expression as a sum of cubes. In this article, we will answer some frequently asked questions about sum of cubes.

Q&A

Q: What is a sum of cubes?

A: A sum of cubes is a mathematical expression that can be factored into the product of three binomials. It is a fundamental concept in algebra and is used to simplify complex expressions.

Q: How do I write an expression as a sum of cubes?

A: To write an expression as a sum of cubes, you need to identify the values of $a$ and $b$ in the general form $a^3 + b^3$. You can then use this information to factor the expression into the product of three binomials.

Q: What is the general form of a sum of cubes?

A: The general form of a sum of cubes is $a^3 + b^3$, where $a$ and $b$ are any real numbers.

Q: Can I write any expression as a sum of cubes?

A: No, not all expressions can be written as a sum of cubes. However, many expressions can be factored into the product of three binomials using the sum of cubes formula.

Q: How do I factor an expression using the sum of cubes formula?

A: To factor an expression using the sum of cubes formula, you need to identify the values of $a$ and $b$ in the general form $a^3 + b^3$. You can then use this information to factor the expression into the product of three binomials.

Q: What are some common examples of sum of cubes?

A: Some common examples of sum of cubes include:

• $a^3 + b^3$
• $a^3 - b^3$
• $a^3 + 2ab^2$
• $a^3 - 2ab^2$

Q: Can I use the sum of cubes formula to factor expressions with negative numbers?

A: Yes, you can use the sum of cubes formula to factor expressions with negative numbers. However, you need to be careful when working with negative numbers, as the formula may not work as expected.

Q: How do I simplify expressions using the sum of cubes formula?

A: To simplify expressions using the sum of cubes formula, you need to factor the expression into the product of three binomials. You can then use algebraic manipulations to simplify the expression.

Conclusion

In this article, we have answered some frequently asked questions about sum of cubes. We have explored the general form of a sum of cubes, how to write an expression as a sum of cubes, and how to factor expressions using the sum of cubes formula. We have also provided some common examples of sum of cubes and discussed how to simplify expressions using the sum of cubes formula.

Final Answer

The final answer is that sum of cubes is a fundamental concept in algebra that can be used to simplify complex expressions. By understanding the general form of a sum of cubes and how to factor expressions using the sum of cubes formula, you can simplify expressions and solve problems more efficiently.

Additional Resources

• Sum of Cubes Formula
• Algebraic Manipulations
• Factoring Expressions

Final Tips

• Practice, practice, practice! The more you practice factoring expressions using the sum of cubes formula, the more comfortable you will become with the concept.
• Use algebraic manipulations to simplify expressions and solve problems more efficiently.
• Don't be afraid to ask for help if you are struggling with a problem or concept.