Applying the Distributive Property and Order of Operations

Oct 13, 2025 by talynt.org 79 views

Introduction

When it comes to simplifying algebraic expressions, there are several rules and properties that we need to follow. One of the most important properties is the distributive property, which allows us to expand expressions with parentheses. In this article, we will explore the distributive property and how it can be used to simplify expressions. We will also discuss the order of operations, which is a set of rules that tells us which operations to perform first when simplifying expressions.

The Distributive Property

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

a(b + c) = ab + ac

This means that we can distribute the multiplication operation to each term inside the parentheses. For example, consider the expression 2(x + 3). Using the distributive property, we can expand this expression as follows:

2(x + 3) = 2x + 6

As we can see, the distributive property allows us to simplify expressions by distributing the multiplication operation to each term inside the parentheses.

Order of Operations

When simplifying expressions, we need to follow a set of rules known as the order of operations. The order of operations is a set of rules that tells us which operations to perform first when simplifying expressions. The order of operations is as follows:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Applying the Distributive Property and Order of Operations

Now that we have discussed the distributive property and the order of operations, let's apply these concepts to the expression given in the problem:

$3(-4.5b - 2.1) - (6b + 0.6)$

Using the distributive property, we can expand the first term as follows:

$3(-4.5b - 2.1) = -13.5b - 6.3$

Now, let's simplify the expression by applying the order of operations:

$-13.5b - 6.3 - (6b + 0.6)$

Using the distributive property again, we can expand the second term as follows:

$-13.5b - 6.3 - 6b - 0.6$

Now, let's combine like terms:

$-13.5b - 6b - 6.3 - 0.6$

Combining the like terms, we get:

$-19.5b - 6.9$

Conclusion

In this article, we have discussed the distributive property and the order of operations, and how they can be used to simplify algebraic expressions. We have applied these concepts to the expression given in the problem and have shown that the correct answer is $-19.5b - 6.9$ . We hope that this article has provided a clear and concise explanation of the distributive property and the order of operations, and how they can be used to simplify algebraic expressions.

Final Answer

The final answer to the problem is:

C. $-19.5b - 6.9$

Introduction

The Distributive Property

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

a(b + c) = ab + ac

2(x + 3) = 2x + 6

As we can see, the distributive property allows us to simplify expressions by distributing the multiplication operation to each term inside the parentheses.

Order of Operations

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Applying the Distributive Property and Order of Operations

Now that we have discussed the distributive property and the order of operations, let's apply these concepts to the expression given in the problem:

$3(-4.5b - 2.1) - (6b + 0.6)$

Using the distributive property, we can expand the first term as follows:

$3(-4.5b - 2.1) = -13.5b - 6.3$

Now, let's simplify the expression by applying the order of operations:

$-13.5b - 6.3 - (6b + 0.6)$

Using the distributive property again, we can expand the second term as follows:

$-13.5b - 6.3 - 6b - 0.6$

Now, let's combine like terms:

$-13.5b - 6b - 6.3 - 0.6$

Combining the like terms, we get:

$-19.5b - 6.9$

Q&A

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions with parentheses. It states that for any 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, simply multiply the number outside the parentheses to each term inside the parentheses.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when simplifying expressions. The order of operations is as follows:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I apply the order of operations?

A: To apply the order of operations, simply follow the order of operations rules. First, evaluate any expressions inside parentheses. Next, evaluate any exponential expressions. Then, evaluate any multiplication and division operations from left to right. Finally, evaluate any addition and subtraction operations from left to right.

Q: Can I use the distributive property and order of operations together?

A: Yes, you can use the distributive property and order of operations together. In fact, they are often used together to simplify complex expressions.

Q: What is the final answer to the problem?

A: The final answer to the problem is:

C. $-19.5b - 6.9$

Conclusion

Final Answer

The final answer to the problem is:

C. $-19.5b - 6.9$