Simplify The Expression:${ 3(2+x) + 4(3+4x) }$

Mar 11, 2025 by ADMIN 48 views

Simplify the Expression: $3(2+x) + 4(3+4x)$

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Introduction

In this article, we will simplify the given algebraic expression $3(2+x) + 4(3+4x)$ . Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. The given expression involves the distributive property, which is a crucial concept in algebra. By simplifying this expression, we will demonstrate the application of the distributive property and other algebraic rules.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand expressions involving multiplication and addition. It states that for any real numbers $a$ , $b$ , and $c$ , the following equation holds:

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

This property can be applied to simplify expressions involving multiple terms. In the given expression, we have two terms: $3(2+x)$ and $4(3+4x)$ . We will apply the distributive property to each term separately.

Applying the Distributive Property

Let's start by applying the distributive property to the first term, $3(2+x)$ . We can rewrite this term as:

$3(2+x) = 3 \cdot 2 + 3 \cdot x$

Using the distributive property, we can simplify this expression as:

$3(2+x) = 6 + 3x$

Now, let's apply the distributive property to the second term, $4(3+4x)$ . We can rewrite this term as:

$4(3+4x) = 4 \cdot 3 + 4 \cdot 4x$

Using the distributive property, we can simplify this expression as:

$4(3+4x) = 12 + 16x$

Combining Like Terms

Now that we have simplified both terms, we can combine them to get the final expression. We can rewrite the original expression as:

$3(2+x) + 4(3+4x) = (6 + 3x) + (12 + 16x)$

Using the commutative property of addition, we can rearrange the terms as:

$(6 + 3x) + (12 + 16x) = 6 + 12 + 3x + 16x$

Combining like terms, we get:

$6 + 12 + 3x + 16x = 18 + 19x$

Conclusion

In this article, we simplified the given algebraic expression $3(2+x) + 4(3+4x)$ using the distributive property and other algebraic rules. We demonstrated the application of the distributive property to expand expressions involving multiplication and addition. By simplifying this expression, we showed how to combine like terms and arrive at the final result. This article provides a clear and concise explanation of the steps involved in simplifying algebraic expressions, making it an essential resource for students and professionals alike.

Frequently Asked Questions

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions involving multiplication and addition. It states that for any real numbers $a$ , $b$ , and $c$ , the following equation holds:

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

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply each term inside the parentheses by the factor outside the parentheses. For example, in the expression $3(2+x)$ , you would multiply $3$ by each term inside the parentheses: $3 \cdot 2$ and $3 \cdot x$ .

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

A: The distributive property allows us to expand expressions involving multiplication and addition, while the commutative property allows us to rearrange the terms in an expression without changing its value. For example, in the expression $(6 + 3x) + (12 + 16x)$ , we can rearrange the terms using the commutative property to get $6 + 12 + 3x + 16x$ .

Additional Resources

Step-by-Step Solution

Apply the distributive property to the first term, $3(2+x)$ .
Simplify the first term to get $6 + 3x$ .
Apply the distributive property to the second term, $4(3+4x)$ .
Simplify the second term to get $12 + 16x$ .
Combine the two simplified terms to get the final expression.
Use the commutative property to rearrange the terms in the final expression.
Combine like terms to get the final result.

Example Problems

Simplify the expression $2(3+x) + 4(2+3x)$ .
Simplify the expression $3(2-x) + 2(1+2x)$ .

Practice Problems

Simplify the expression $4(2+x) + 3(1+2x)$ .
Simplify the expression $2(3-x) + 4(2+x)$ .

Final Answer

The final answer is $\boxed{18 + 19x}$ .

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Introduction

In our previous article, we simplified the algebraic expression $3(2+x) + 4(3+4x)$ using the distributive property and other algebraic rules. In this article, we will address some of the most frequently asked questions related to algebraic expression simplification. Whether you're a student struggling with algebra or a professional looking to refresh your skills, this article is designed to provide you with the answers you need to succeed.

Q&A

Q: What is the distributive property, and how do I apply it?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions involving multiplication and addition. To apply the distributive property, you need to multiply each term inside the parentheses by the factor outside the parentheses. For example, in the expression $3(2+x)$ , you would multiply $3$ by each term inside the parentheses: $3 \cdot 2$ and $3 \cdot x$ .

Q: How do I simplify expressions involving multiple terms?

A: To simplify expressions involving multiple terms, you need to apply the distributive property to each term separately. Then, you can combine like terms to get the final result. For example, in the expression $3(2+x) + 4(3+4x)$ , you would apply the distributive property to each term separately, and then combine like terms to get the final result.

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

Q: How do I handle expressions with negative coefficients?

A: When working with expressions that have negative coefficients, you need to apply the distributive property and then combine like terms. For example, in the expression $-3(2+x)$ , you would apply the distributive property to get $-6 - 3x$ , and then combine like terms to get the final result.

Q: Can I simplify expressions with variables in the denominator?

A: Yes, you can simplify expressions with variables in the denominator. However, you need to be careful when working with fractions and variables. For example, in the expression $\frac{3(2+x)}{2}$ , you would apply the distributive property to get $\frac{6 + 3x}{2}$ , and then simplify the fraction to get the final result.

Common Mistakes

Mistake 1: Failing to apply the distributive property

Solution: Make sure to apply the distributive property to each term separately.
Example: Simplify the expression $3(2+x) + 4(3+4x)$ by applying the distributive property to each term separately.

Mistake 2: Not combining like terms

Solution: Make sure to combine like terms after applying the distributive property.
Example: Simplify the expression $6 + 12 + 3x + 16x$ by combining like terms.

Mistake 3: Not checking for negative coefficients

Solution: Make sure to check for negative coefficients when working with expressions.
Example: Simplify the expression $-3(2+x)$ by applying the distributive property and then combining like terms.

Tips and Tricks

Tip 1: Use the distributive property to simplify expressions

Solution: Apply the distributive property to each term separately to simplify expressions.
Example: Simplify the expression $3(2+x) + 4(3+4x)$ by applying the distributive property to each term separately.

Tip 2: Combine like terms to simplify expressions

Solution: Combine like terms after applying the distributive property to simplify expressions.
Example: Simplify the expression $6 + 12 + 3x + 16x$ by combining like terms.

Tip 3: Check for negative coefficients when working with expressions

Solution: Make sure to check for negative coefficients when working with expressions.
Example: Simplify the expression $-3(2+x)$ by applying the distributive property and then combining like terms.

Conclusion

In this article, we addressed some of the most frequently asked questions related to algebraic expression simplification. Whether you're a student struggling with algebra or a professional looking to refresh your skills, this article is designed to provide you with the answers you need to succeed. By following the tips and tricks outlined in this article, you can simplify expressions with ease and become a master of algebraic expression simplification.

Frequently Asked Questions

Q: What is the distributive property, and how do I apply it?

Q: How do I simplify expressions involving multiple terms?

A: To simplify expressions involving multiple terms, you need to apply the distributive property to each term separately. Then, you can combine like terms to get the final result.

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

Additional Resources

Step-by-Step Solution

Apply the distributive property to each term separately.
Combine like terms to get the final result.
Check for negative coefficients when working with expressions.

Example Problems

Simplify the expression $2(3+x) + 4(2+3x)$ .
Simplify the expression $3(2-x) + 2(1+2x)$ .

Practice Problems

Simplify the expression $4(2+x) + 3(1+2x)$ .
Simplify the expression $2(3-x) + 4(2+x)$ .

Final Answer

The final answer is $\boxed{18 + 19x}$ .