What Is The Factored Form Of The Expression $27m^3 - 64n^3$?A. $(3m - 4n)(9m^2 + 12mn + 16n^2)$B. $ ( 3 M + 4 N ) ( 9 M 2 − 12 M N + 16 N 2 ) (3m + 4n)(9m^2 - 12mn + 16n^2) ( 3 M + 4 N ) ( 9 M 2 − 12 Mn + 16 N 2 ) [/tex]C. $(4m - 3n)(16m^2 + 12mn + 9n^2)$D. $(4m + 3n)(16m^2 - 12mn

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Understanding the Problem

The given expression is a difference of cubes, which can be factored using the formula $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. In this case, we have $27m^3 - 64n^3$, where $a = 3m$ and $b = 4n$. Our goal is to factor this expression and determine the correct form among the given options.

Factoring the Expression

To factor the expression, we can use the formula for the difference of cubes. We start by identifying the values of $a$ and $b$, which are $3m$ and $4n$, respectively. Then, we can apply the formula to obtain the factored form.

$$27m^3 - 64n^3 = (3m)^3 - (4n)^3$$

Using the formula, we can rewrite the expression as:

$$(3m)^3 - (4n)^3 = (3m - 4n)((3m)^2 + (3m)(4n) + (4n)^2)$$

Simplifying the expression, we get:

$$(3m - 4n)(9m^2 + 12mn + 16n^2)$$

Evaluating the Options

Now that we have factored the expression, we can compare it with the given options to determine the correct form.

Option A: $(3m - 4n)(9m^2 + 12mn + 16n^2)$

This option matches the factored form we obtained earlier.

Option B: $(3m + 4n)(9m^2 - 12mn + 16n^2)$

This option does not match the factored form we obtained earlier.

Option C: $(4m - 3n)(16m^2 + 12mn + 9n^2)$

This option does not match the factored form we obtained earlier.

Option D: $(4m + 3n)(16m^2 - 12mn + 9n^2)$

This option does not match the factored form we obtained earlier.

Conclusion

Based on our analysis, the correct factored form of the expression $27m^3 - 64n^3$ is:

$$(3m - 4n)(9m^2 + 12mn + 16n^2)$$

This matches option A, which is the correct answer.

Key Takeaways

• The expression $27m^3 - 64n^3$ can be factored using the formula for the difference of cubes.
• The factored form of the expression is $(3m - 4n)(9m^2 + 12mn + 16n^2)$.
• This matches option A, which is the correct answer.

Additional Examples

• Factoring the expression $8x^3 - 27y^3$ using the formula for the difference of cubes.
• Factoring the expression $125a^3 - 216b^3$ using the formula for the difference of cubes.

Real-World Applications

• Factoring expressions is an important skill in algebra and is used in a variety of real-world applications, such as:
• Solving systems of equations
• Finding the roots of a polynomial equation
• Factoring quadratic expressions

Common Mistakes

• Failing to identify the values of $a$ and $b$ in the expression.
• Failing to apply the formula for the difference of cubes correctly.
• Failing to simplify the expression correctly.

Tips and Tricks

• Make sure to identify the values of $a$ and $b$ in the expression before applying the formula.
• Use the formula for the difference of cubes to factor the expression.
• Simplify the expression correctly to obtain the factored form.

Conclusion

Factoring the expression $27m^3 - 64n^3$ using the formula for the difference of cubes is an important skill in algebra. By following the steps outlined in this article, you can factor the expression and determine the correct form among the given options.

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Frequently Asked Questions

Q: What is the formula for factoring the difference of cubes?

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

Q: How do I identify the values of $a$ and $b$ in the expression?

A: To identify the values of $a$ and $b$, you need to look for the two terms that are being subtracted. In this case, the two terms are $27m^3$ and $64n^3$. Therefore, $a = 3m$ and $b = 4n$.

Q: How do I apply the formula for the difference of cubes?

A: To apply the formula, you need to substitute the values of $a$ and $b$ into the formula. In this case, you would substitute $a = 3m$ and $b = 4n$ into the formula $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

Q: What is the factored form of the expression $27m^3 - 64n^3$?

A: The factored form of the expression $27m^3 - 64n^3$ is $(3m - 4n)(9m^2 + 12mn + 16n^2)$.

Q: How do I simplify the expression after factoring?

A: To simplify the expression, you need to multiply the two binomials together. In this case, you would multiply $(3m - 4n)$ and $(9m^2 + 12mn + 16n^2)$ together.

Q: What are some common mistakes to avoid when factoring the difference of cubes?

A: Some common mistakes to avoid when factoring the difference of cubes include:

• Failing to identify the values of $a$ and $b$ in the expression.
• Failing to apply the formula for the difference of cubes correctly.
• Failing to simplify the expression correctly.

Q: What are some tips and tricks for factoring the difference of cubes?

A: Some tips and tricks for factoring the difference of cubes include:

• Make sure to identify the values of $a$ and $b$ in the expression before applying the formula.
• Use the formula for the difference of cubes to factor the expression.
• Simplify the expression correctly to obtain the factored form.

Q: How do I use the factored form of the expression in real-world applications?

A: The factored form of the expression can be used in a variety of real-world applications, such as:

• Solving systems of equations
• Finding the roots of a polynomial equation
• Factoring quadratic expressions

Q: What are some additional examples of factoring the difference of cubes?

A: Some additional examples of factoring the difference of cubes include:

• Factoring the expression $8x^3 - 27y^3$ using the formula for the difference of cubes.
• Factoring the expression $125a^3 - 216b^3$ using the formula for the difference of cubes.

Conclusion

Factoring the expression $27m^3 - 64n^3$ using the formula for the difference of cubes is an important skill in algebra. By following the steps outlined in this article, you can factor the expression and determine the correct form among the given options.