Choose The Properties That Can Be Used To Rewrite $2(a+5)-4$ As $2a+6$.$2(a+5)-4=2 \cdot A+2 \cdot 5-4$A. Distributive Property B. Commutative Property C. Associative Property

Introduction

Algebraic expressions are a fundamental concept in mathematics, and rewriting them is an essential skill for students and professionals alike. In this article, we will explore the properties that can be used to rewrite algebraic expressions, with a focus on the distributive, commutative, and associative properties. We will examine the expression $2(a+5)-4$ and rewrite it as $2a+6$, highlighting the properties used in the process.

The Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses by the factor outside the parentheses. The distributive property is represented by the following equation:

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

In the expression $2(a+5)-4$, we can use the distributive property to expand the expression inside the parentheses:

$$2(a+5) = 2a + 2 \cdot 5$$

Using the distributive property, we can rewrite the expression as:

$$2(a+5)-4 = 2a + 10 - 4$$

The Commutative Property

The commutative property is a property of addition and multiplication that states that the order of the terms does not change the result. The commutative property is represented by the following equations:

$$a + b = b + a$$

$$a \cdot b = b \cdot a$$

In the expression $2a + 10 - 4$, we can use the commutative property to rearrange the terms:

$$2a + 10 - 4 = 2a + (10 - 4)$$

Using the commutative property, we can rewrite the expression as:

$$2a + 10 - 4 = 2a + 6$$

The Associative Property

The associative property is a property of addition and multiplication that states that the order in which we perform the operations does not change the result. The associative property is represented by the following equations:

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

$$(a \cdot b) \cdot c = a \cdot (b \cdot c)$$

In the expression $2a + 6$, we can use the associative property to rearrange the terms:

$$(2a + 6) = 2a + 6$$

Using the associative property, we can rewrite the expression as:

$$(2a + 6) = 2a + 6$$

Conclusion

In conclusion, the distributive, commutative, and associative properties are essential tools for rewriting algebraic expressions. By using these properties, we can simplify complex expressions and make them easier to understand. In this article, we have explored the properties used to rewrite the expression $2(a+5)-4$ as $2a+6$, highlighting the importance of each property in the process.

Choosing the Right Properties

When rewriting algebraic expressions, it is essential to choose the right properties to use. Here are some tips to help you choose the right properties:

• Use the distributive property to expand expressions: When you see an expression inside parentheses, use the distributive property to expand it.
• Use the commutative property to rearrange terms: When you see a group of terms, use the commutative property to rearrange them.
• Use the associative property to rearrange operations: When you see a group of operations, use the associative property to rearrange them.

By following these tips, you can choose the right properties to use when rewriting algebraic expressions.

Examples and Practice

Here are some examples and practice problems to help you reinforce your understanding of the distributive, commutative, and associative properties:

• Example 1: Rewrite the expression $3(x+2)-5$ using the distributive property.
• Example 2: Rewrite the expression $2(a+3)+4$ using the commutative property.
• Example 3: Rewrite the expression $(a+2)+3$ using the associative property.

Practice problems:

• Rewrite the expression $4(x+1)-2$ using the distributive property.
• Rewrite the expression $3(a+2)+1$ using the commutative property.
• Rewrite the expression $(a+1)+2$ using the associative property.

Conclusion

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses by the factor outside the parentheses. The distributive property is represented by the following equation:

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

Q: How do I use the distributive property to rewrite an expression?

A: To use the distributive property to rewrite an expression, follow these steps:

1. Identify the expression inside the parentheses.
2. Multiply each term inside the parentheses by the factor outside the parentheses.
3. Simplify the resulting expression.

Q: What is the commutative property?

A: The commutative property is a property of addition and multiplication that states that the order of the terms does not change the result. The commutative property is represented by the following equations:

$$a + b = b + a$$

$$a \cdot b = b \cdot a$$

Q: How do I use the commutative property to rewrite an expression?

A: To use the commutative property to rewrite an expression, follow these steps:

1. Identify the expression that can be rearranged using the commutative property.
2. Rearrange the terms to change the order of the terms.
3. Simplify the resulting expression.

Q: What is the associative property?

A: The associative property is a property of addition and multiplication that states that the order in which we perform the operations does not change the result. The associative property is represented by the following equations:

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

$$(a \cdot b) \cdot c = a \cdot (b \cdot c)$$

Q: How do I use the associative property to rewrite an expression?

A: To use the associative property to rewrite an expression, follow these steps:

1. Identify the expression that can be rearranged using the associative property.
2. Rearrange the operations to change the order of the operations.
3. Simplify the resulting expression.

Q: Can I use the distributive, commutative, and associative properties together?

A: Yes, you can use the distributive, commutative, and associative properties together to rewrite an expression. For example, you can use the distributive property to expand an expression, and then use the commutative property to rearrange the terms.

Q: Are the distributive, commutative, and associative properties only used in algebra?

A: No, the distributive, commutative, and associative properties are used in many areas of mathematics, including arithmetic, geometry, and calculus.

Q: Can I use the distributive, commutative, and associative properties to solve word problems?

A: Yes, you can use the distributive, commutative, and associative properties to solve word problems. For example, if you are given a problem that involves adding or multiplying numbers, you can use the distributive, commutative, and associative properties to simplify the expression and find the solution.

Q: Are the distributive, commutative, and associative properties important in real-life applications?

A: Yes, the distributive, commutative, and associative properties are important in many real-life applications, including finance, engineering, and science. For example, in finance, the distributive property is used to calculate interest rates and investments. In engineering, the commutative property is used to design and build complex systems. In science, the associative property is used to analyze and model complex phenomena.

Conclusion

In conclusion, the distributive, commutative, and associative properties are essential tools for rewriting algebraic expressions and solving mathematical problems. By understanding and using these properties, you can simplify complex expressions and find solutions to mathematical problems.