Find The Product: $2x(x - 8n$\]

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Introduction

In algebra, the product of two or more expressions is a fundamental concept that is used to simplify complex equations and expressions. In this article, we will focus on finding the product of two expressions, specifically $2x(x - 8n)$, and explore the various methods and techniques used to simplify and expand this expression.

Understanding the Expression

The given expression is $2x(x - 8n)$. To find the product, we need to multiply the two expressions together. The first expression is $2x$, and the second expression is $(x - 8n)$. We will use the distributive property to multiply these two expressions.

Using the Distributive Property

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

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

We can use this property to multiply the two expressions together. We will multiply the first expression, $2x$, by each term in the second expression, $(x - 8n)$.

Step 1: Multiply $2x$ by $x$

To multiply $2x$ by $x$, we can use the commutative property of multiplication, which states that the order of the factors does not change the product. Therefore, we can write:

$$2x \cdot x = x \cdot 2x = 2x^2$$

Step 2: Multiply $2x$ by $-8n$

To multiply $2x$ by $-8n$, we can use the distributive property again. We will multiply $2x$ by each term in the second expression, $-8n$.

$$2x \cdot -8n = -16xn$$

Combining the Results

Now that we have multiplied each term in the second expression by $2x$, we can combine the results to find the product of the two expressions.

$$2x(x - 8n) = 2x^2 - 16xn$$

Simplifying the Expression

The expression $2x^2 - 16xn$ is the simplified form of the product of the two expressions. We can further simplify this expression by factoring out the greatest common factor (GCF).

Factoring Out the GCF

The GCF of $2x^2$ and $-16xn$ is $2x$. We can factor out $2x$ from each term in the expression.

$$2x^2 - 16xn = 2x(x - 8n)$$

Conclusion

In this article, we have explored the concept of finding the product of two expressions, specifically $2x(x - 8n)$. We have used the distributive property to multiply the two expressions together and have simplified the resulting expression by factoring out the greatest common factor. The final answer is $2x(x - 8n) = 2x^2 - 16xn$.

Common Mistakes to Avoid

When finding the product of two expressions, it is easy to make mistakes. Here are some common mistakes to avoid:

• Not using the distributive property: Failing to use the distributive property can lead to incorrect results.
• Not simplifying the expression: Failing to simplify the expression can lead to a more complex and difficult-to-read result.
• Not factoring out the GCF: Failing to factor out the GCF can lead to a more complex and difficult-to-read result.

Real-World Applications

The concept of finding the product of two expressions has many real-world applications. Here are a few examples:

• Algebraic equations: The product of two expressions is used to solve algebraic equations.
• Calculus: The product of two expressions is used to find the derivative of a function.
• Physics: The product of two expressions is used to find the force exerted on an object.

Final Thoughts

Introduction

In our previous article, we explored the concept of finding the product of two expressions, specifically $2x(x - 8n)$. We used the distributive property to multiply the two expressions together and simplified the resulting expression by factoring out the greatest common factor. In this article, we will answer some common questions related to finding the product of two expressions.

Q: What is the distributive property?

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

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

This property allows us to multiply a single expression by each term in a binomial expression.

Q: How do I use the distributive property to find the product of two expressions?

A: To use the distributive property to find the product of two expressions, follow these steps:

1. Multiply the first expression by each term in the second expression.
2. Simplify the resulting expression by combining like terms.
3. Factor out the greatest common factor (GCF) to simplify the expression further.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest expression that divides each term in an expression without leaving a remainder. For example, in the expression $2x^2 - 16xn$, the GCF is $2x$.

Q: How do I factor out the GCF?

A: To factor out the GCF, follow these steps:

1. Identify the GCF of the expression.
2. Divide each term in the expression by the GCF.
3. Write the resulting expression as a product of the GCF and the remaining expression.

Q: What are some common mistakes to avoid when finding the product of two expressions?

A: Some common mistakes to avoid when finding the product of two expressions include:

• Not using the distributive property
• Not simplifying the expression
• Not factoring out the GCF

Q: How do I apply the concept of finding the product of two expressions to real-world problems?

A: The concept of finding the product of two expressions has many real-world applications, including:

• Algebraic equations
• Calculus
• Physics

For example, in physics, the product of two expressions is used to find the force exerted on an object.

Q: What are some tips for simplifying expressions?

A: Some tips for simplifying expressions include:

• Combining like terms
• Factoring out the GCF
• Using the distributive property

Q: How do I check my work when finding the product of two expressions?

A: To check your work when finding the product of two expressions, follow these steps:

1. Multiply the two expressions together.
2. Simplify the resulting expression.
3. Check that the expression is in its simplest form.

Conclusion

In conclusion, finding the product of two expressions is a fundamental concept in algebra that has many real-world applications. By using the distributive property and simplifying the resulting expression, we can find the product of two expressions and solve complex equations and problems. We hope that this Q&A article has provided you with a better understanding of the concept and has helped you to avoid common mistakes.