Simplify The Expression: $ (4n - 4)(3n + 3) $

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Introduction

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In algebra, simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements. One of the most common techniques used to simplify expressions is the distributive property, which allows us to expand the product of two or more terms. In this article, we will use the distributive property to simplify the expression $(4n - 4)(3n + 3)$.

The Distributive Property

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The distributive property is a fundamental concept in algebra that states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$$a(b + c) = ab + ac$$

This property can be extended to more than two terms, and it is a powerful tool for simplifying expressions. In the case of the expression $(4n - 4)(3n + 3)$, we can use the distributive property to expand the product of the two terms.

Expanding the Product

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To expand the product of the two terms, we need to multiply each term in the first expression by each term in the second expression. This can be done using the distributive property, which states that:

$$(a + b)(c + d) = ac + ad + bc + bd$$

Using this property, we can expand the product of the two terms as follows:

$$(4n - 4)(3n + 3) = (4n)(3n) + (4n)(3) + (-4)(3n) + (-4)(3)$$

Simplifying the Expression

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Now that we have expanded the product of the two terms, we can simplify the expression by combining like terms. This involves combining the terms that have the same variable and coefficient.

$$(4n)(3n) = 12n^2$$

$$(4n)(3) = 12n$$

$$(-4)(3n) = -12n$$

$$(-4)(3) = -12$$

Combining Like Terms

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Now that we have simplified each term, we can combine like terms to get the final expression. This involves adding or subtracting the terms that have the same variable and coefficient.

$$12n^2 + 12n - 12n - 12$$

Final Expression

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After combining like terms, we get the final expression:

$$12n^2 - 12$$

Conclusion

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In this article, we used the distributive property to simplify the expression $(4n - 4)(3n + 3)$. We expanded the product of the two terms using the distributive property and then simplified the expression by combining like terms. The final expression is $12n^2 - 12$.

Example Use Case

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The expression $(4n - 4)(3n + 3)$ can be used in a variety of mathematical contexts, such as solving equations and manipulating algebraic expressions. For example, if we want to solve the equation $x = (4n - 4)(3n + 3)$, we can use the simplified expression $12n^2 - 12$ to find the value of $x$.

Tips and Tricks

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When simplifying expressions using the distributive property, it's essential to remember the following tips and tricks:

• Use the distributive property to expand the product of two or more terms.
• Simplify each term by combining like terms.
• Combine like terms to get the final expression.
• Check your work by plugging in values for the variables.

Common Mistakes

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When simplifying expressions using the distributive property, it's easy to make mistakes. Here are some common mistakes to avoid:

• Forgetting to use the distributive property to expand the product of two or more terms.
• Not simplifying each term by combining like terms.
• Not combining like terms to get the final expression.
• Not checking your work by plugging in values for the variables.

Final Thoughts

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Simplifying expressions using the distributive property is a crucial skill that helps us solve equations and manipulate mathematical statements. By following the steps outlined in this article, you can simplify expressions like $(4n - 4)(3n + 3)$ and get the final expression $12n^2 - 12$. Remember to use the distributive property to expand the product of two or more terms, simplify each term by combining like terms, and combine like terms to get the final expression. With practice and patience, you can become proficient in simplifying expressions using the distributive property.

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Introduction

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In our previous article, we used the distributive property to simplify the expression $(4n - 4)(3n + 3)$. We expanded the product of the two terms and then simplified the expression by combining like terms. In this article, we will answer some frequently asked questions about simplifying expressions using the distributive property.

Q&A

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Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$$a(b + c) = ab + ac$$

This property can be extended to more than two terms, and it is a powerful tool for simplifying expressions.

Q: How do I use the distributive property to simplify expressions?

A: To use the distributive property to simplify expressions, you need to follow these steps:

1. Expand the product of two or more terms using the distributive property.
2. Simplify each term by combining like terms.
3. Combine like terms to get the final expression.

Q: What is the difference between the distributive property and the commutative property?

A: The distributive property and the commutative property are two different properties in algebra. The distributive property states that:

$$a(b + c) = ab + ac$$

The commutative property states that:

$$a + b = b + a$$

The distributive property is used to expand the product of two or more terms, while the commutative property is used to rearrange the order of terms.

Q: Can I use the distributive property to simplify expressions with variables and constants?

A: Yes, you can use the distributive property to simplify expressions with variables and constants. For example, if you have the expression $(2x + 3)(4x + 5)$, you can use the distributive property to expand the product of the two terms.

Q: How do I know when to use the distributive property to simplify expressions?

A: You should use the distributive property to simplify expressions when you have a product of two or more terms. For example, if you have the expression $(x + 2)(x + 3)$, you can use the distributive property to expand the product of the two terms.

Q: Can I use the distributive property to simplify expressions with fractions?

A: Yes, you can use the distributive property to simplify expressions with fractions. For example, if you have the expression $(\frac{1}{2}x + \frac{1}{3})(\frac{2}{3}x + \frac{1}{2})$, you can use the distributive property to expand the product of the two terms.

Q: How do I check my work when simplifying expressions using the distributive property?

A: To check your work when simplifying expressions using the distributive property, you should:

1. Plug in values for the variables to see if the expression simplifies to the expected value.
2. Use a calculator to evaluate the expression and see if it simplifies to the expected value.
3. Check your work by comparing it to the original expression.

Example Use Cases

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The distributive property can be used in a variety of mathematical contexts, such as solving equations and manipulating algebraic expressions. Here are some example use cases:

• Solving equations: The distributive property can be used to solve equations by expanding the product of two or more terms.
• Manipulating algebraic expressions: The distributive property can be used to simplify algebraic expressions by expanding the product of two or more terms.
• Calculus: The distributive property can be used in calculus to simplify expressions and solve equations.

Tips and Tricks

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When simplifying expressions using the distributive property, it's essential to remember the following tips and tricks:

• Use the distributive property to expand the product of two or more terms.
• Simplify each term by combining like terms.
• Combine like terms to get the final expression.
• Check your work by plugging in values for the variables.

Common Mistakes

---

When simplifying expressions using the distributive property, it's easy to make mistakes. Here are some common mistakes to avoid:

• Forgetting to use the distributive property to expand the product of two or more terms.
• Not simplifying each term by combining like terms.
• Not combining like terms to get the final expression.
• Not checking your work by plugging in values for the variables.

Final Thoughts

---

Simplifying expressions using the distributive property is a crucial skill that helps us solve equations and manipulate mathematical statements. By following the steps outlined in this article, you can simplify expressions like $(4n - 4)(3n + 3)$ and get the final expression $12n^2 - 12$. Remember to use the distributive property to expand the product of two or more terms, simplify each term by combining like terms, and combine like terms to get the final expression. With practice and patience, you can become proficient in simplifying expressions using the distributive property.