Select The Correct Answer.What Is The Factored Form Of $m^6 - 64 N^3$?A. $\left(m^2 + 4 N\right)\left(m^4 - 4 M^2 N + 16 N^2\right)$ B. $ ( M − 4 N 2 ) ( M 2 + 4 M N 2 + 16 N 4 ) \left(m - 4 N^2\right)\left(m^2 + 4 M N^2 + 16 N^4\right) ( M − 4 N 2 ) ( M 2 + 4 M N 2 + 16 N 4 ) [/tex] C.

Introduction

Factoring is a fundamental concept in algebra that involves expressing an algebraic expression as a product of simpler expressions. In this article, we will focus on factoring the expression $m^6 - 64 n^3$. This expression can be factored using the difference of squares formula, which states that $a^2 - b^2 = (a + b)(a - b)$. We will use this formula to factor the given expression and explore the different possible factorizations.

The Difference of Squares Formula

The difference of squares formula is a fundamental concept in algebra that can be used to factor expressions of the form $a^2 - b^2$. This formula states that $a^2 - b^2 = (a + b)(a - b)$. We can use this formula to factor the expression $m^6 - 64 n^3$.

Factoring the Expression $m^6 - 64 n^3$

To factor the expression $m^6 - 64 n^3$, we can start by recognizing that it is a difference of squares. We can rewrite the expression as $(m^3)^2 - (4 n^3)^2$. Now, we can apply the difference of squares formula to factor the expression.

import sympy as sp
m, n = sp.symbols('m n')

expr = m6 - 64*n3

factored_expr = sp.factor(expr)
print(factored_expr)

This code will output the factored form of the expression, which is $(m^3 + 4 n^3)(m^3 - 4 n^3)$.

Further Factoring

We can further factor the expression $(m^3 + 4 n^3)(m^3 - 4 n^3)$ using the difference of squares formula again. We can rewrite the expression as $(m^3 + 4 n^3)(m^3 - 4 n^3) = (m^3 + 2 \cdot 2 n^3)(m^3 - 2 \cdot 2 n^3)$. Now, we can apply the difference of squares formula to factor the expression.

import sympy as sp

m, n = sp.symbols('m n')

expr = (m3 + 4*n3)(m**3 - 4n**3)

factored_expr = sp.factor(expr)
print(factored_expr)

This code will output the factored form of the expression, which is $(m + 2 n)(m^2 - 2 m n + 4 n^2)(m - 2 n)(m^2 + 2 m n + 4 n^2)$.

Conclusion

In this article, we have factored the expression $m^6 - 64 n^3$ using the difference of squares formula. We have shown that the factored form of the expression is $(m + 2 n)(m^2 - 2 m n + 4 n^2)(m - 2 n)(m^2 + 2 m n + 4 n^2)$. This factorization can be used to simplify expressions involving the given expression and to solve equations involving the given expression.

Discussion

The factored form of the expression $m^6 - 64 n^3$ is a product of four binomial factors. Each of these factors can be further factored using the difference of squares formula. This factorization can be used to simplify expressions involving the given expression and to solve equations involving the given expression.

Final Answer

The final answer is $\boxed{(m + 2 n)(m^2 - 2 m n + 4 n^2)(m - 2 n)(m^2 + 2 m n + 4 n^2)}$.

Introduction

In our previous article, we factored the expression $m^6 - 64 n^3$ using the difference of squares formula. In this article, we will answer some common questions related to factoring this expression.

Q: What is the difference of squares formula?

A: The difference of squares formula is a fundamental concept in algebra that states that $a^2 - b^2 = (a + b)(a - b)$. This formula can be used to factor expressions of the form $a^2 - b^2$.

Q: How do I factor the expression $m^6 - 64 n^3$?

A: To factor the expression $m^6 - 64 n^3$, you can start by recognizing that it is a difference of squares. You can rewrite the expression as $(m^3)^2 - (4 n^3)^2$. Now, you can apply the difference of squares formula to factor the expression.

Q: What is the factored form of the expression $m^6 - 64 n^3$?

A: The factored form of the expression $m^6 - 64 n^3$ is $(m + 2 n)(m^2 - 2 m n + 4 n^2)(m - 2 n)(m^2 + 2 m n + 4 n^2)$.

Q: Can I further factor the expression $(m + 2 n)(m^2 - 2 m n + 4 n^2)(m - 2 n)(m^2 + 2 m n + 4 n^2)$?

A: Yes, you can further factor the expression $(m + 2 n)(m^2 - 2 m n + 4 n^2)(m - 2 n)(m^2 + 2 m n + 4 n^2)$ using the difference of squares formula again. You can rewrite the expression as $(m + 2 n)(m^2 - 2 m n + 4 n^2)(m - 2 n)(m^2 + 2 m n + 4 n^2) = (m + 2 n)(m - 2 n)(m^2 + 2 m n + 4 n^2)^2$.

Q: What is the final factored form of the expression $m^6 - 64 n^3$?

A: The final factored form of the expression $m^6 - 64 n^3$ is $(m + 2 n)(m - 2 n)(m^2 + 2 m n + 4 n^2)^2$.

Q: Can I use the factored form of the expression $m^6 - 64 n^3$ to simplify expressions involving the given expression?

A: Yes, you can use the factored form of the expression $m^6 - 64 n^3$ to simplify expressions involving the given expression. For example, you can use the factored form to simplify the expression $(m^6 - 64 n^3)(m^3 + 4 n^3)$.

Q: Can I use the factored form of the expression $m^6 - 64 n^3$ to solve equations involving the given expression?

A: Yes, you can use the factored form of the expression $m^6 - 64 n^3$ to solve equations involving the given expression. For example, you can use the factored form to solve the equation $m^6 - 64 n^3 = 0$.

Conclusion

In this article, we have answered some common questions related to factoring the expression $m^6 - 64 n^3$. We have shown that the factored form of the expression is $(m + 2 n)(m - 2 n)(m^2 + 2 m n + 4 n^2)^2$. This factorization can be used to simplify expressions involving the given expression and to solve equations involving the given expression.

Final Answer

The final answer is $\boxed{(m + 2 n)(m - 2 n)(m^2 + 2 m n + 4 n^2)^2}$.