Simplify The Expression:${ -4(1 - 8n) - 4(8n + 4) }$

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Introduction

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In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. It involves combining like terms, removing unnecessary parentheses, and rearranging the expression to make it easier to work with. In this article, we will focus on simplifying the given expression: $-4(1 - 8n) - 4(8n + 4)$. We will break down the steps involved in simplifying this expression and provide a clear explanation of each step.

Understanding the Expression

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The given expression is a combination of two terms, each of which involves a product of a constant and a polynomial. The first term is $-4(1 - 8n)$, and the second term is $-4(8n + 4)$. To simplify this expression, we need to apply the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.

Step 1: Apply the Distributive Property

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To simplify the expression, we will start by applying the distributive property to each term. For the first term, $-4(1 - 8n)$, we can rewrite it as $-4 \cdot 1 + (-4) \cdot (-8n)$. Similarly, for the second term, $-4(8n + 4)$, we can rewrite it as $-4 \cdot 8n + (-4) \cdot 4$.

Step 3: Simplify the Terms

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Now that we have applied the distributive property, we can simplify each term. For the first term, $-4 \cdot 1 + (-4) \cdot (-8n)$, we can simplify it to $-4 + 32n$. For the second term, $-4 \cdot 8n + (-4) \cdot 4$, we can simplify it to $-32n - 16$.

Step 4: Combine Like Terms

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Now that we have simplified each term, we can combine like terms. The expression $-4 + 32n - 32n - 16$ has two like terms, $32n$ and $-32n$, which cancel each other out. Therefore, the simplified expression is $-4 - 16$.

Step 5: Simplify the Final Expression

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The final expression is $-4 - 16$, which can be simplified to $-20$.

Conclusion

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In this article, we have simplified the given expression $-4(1 - 8n) - 4(8n + 4)$ by applying the distributive property, simplifying the terms, combining like terms, and simplifying the final expression. The simplified expression is $-20$. We hope that this article has provided a clear and step-by-step guide to simplifying the expression.

Frequently Asked Questions

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Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.

Q: How do I simplify an expression?

A: To simplify an expression, you need to apply the distributive property, simplify the terms, combine like terms, and simplify the final expression.

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable and coefficient, while unlike terms are terms that have different variables or coefficients.

Final Answer

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The final answer is $\boxed{-20}$.

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Introduction

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In our previous article, we simplified the expression $-4(1 - 8n) - 4(8n + 4)$ by applying the distributive property, simplifying the terms, combining like terms, and simplifying the final expression. In this article, we will provide a Q&A guide to help you understand the concepts and steps involved in simplifying the expression.

Q&A: Simplifying Expressions

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Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This property allows us to expand a product of a constant and a polynomial.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply the constant by each term inside the parentheses. For example, if you have the expression $-4(1 - 8n)$, you can rewrite it as $-4 \cdot 1 + (-4) \cdot (-8n)$.

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable and coefficient, while unlike terms are terms that have different variables or coefficients. For example, $2x$ and $3x$ are like terms, while $2x$ and $3y$ are unlike terms.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, if you have the expression $2x + 3x$, you can combine the like terms to get $5x$.

Q: What is the final answer to the expression $-4(1 - 8n) - 4(8n + 4)$?

A: The final answer to the expression $-4(1 - 8n) - 4(8n + 4)$ is $-20$.

Q&A: Common Mistakes

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Q: What is the most common mistake when simplifying expressions?

A: The most common mistake when simplifying expressions is not applying the distributive property correctly. This can lead to incorrect simplifications and final answers.

Q: How can I avoid making mistakes when simplifying expressions?

A: To avoid making mistakes when simplifying expressions, you need to carefully apply the distributive property, simplify the terms, combine like terms, and simplify the final expression.

Q: What should I do if I get stuck while simplifying an expression?

A: If you get stuck while simplifying an expression, you should try to break down the expression into smaller parts and simplify each part separately. You can also try to use different methods or approaches to simplify the expression.

Q&A: Advanced Topics

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Q: What is the difference between a polynomial and a rational expression?

A: A polynomial is an expression that consists of variables and coefficients, while a rational expression is an expression that consists of a fraction of polynomials.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and denominator, cancel out any common factors, and simplify the resulting expression.

Q: What is the final answer to the expression $\frac{x^2 + 4x + 4}{x + 2}$?

A: The final answer to the expression $\frac{x^2 + 4x + 4}{x + 2}$ is $x + 2$.

Conclusion

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In this article, we have provided a Q&A guide to help you understand the concepts and steps involved in simplifying expressions. We have covered topics such as the distributive property, like terms, unlike terms, combining like terms, and simplifying rational expressions. We hope that this article has been helpful in clarifying any doubts you may have had about simplifying expressions.

Final Answer

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The final answer is $\boxed{-20}$.