Use The Distributive Property And Combine Like Terms To Simplify: $2p(3q + 12$\].Provide Your Answer Below:

Introduction

Algebraic expressions can be complex and difficult to simplify, but with the right techniques, they can be broken down into manageable parts. In this article, we will explore how to use the distributive property and combine like terms to simplify the expression $2p(3q + 12)$. By following these steps, you will be able to simplify even the most complex algebraic expressions.

Understanding 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 term outside the parentheses. In the expression $2p(3q + 12)$, the distributive property can be applied to expand the expression as follows:

$2p(3q + 12) = 2p \cdot 3q + 2p \cdot 12$

Applying the Distributive Property

Now that we have applied the distributive property, we can simplify the expression further by multiplying each term inside the parentheses by the term outside the parentheses.

$2p \cdot 3q = 6pq$

$2p \cdot 12 = 24p$

Combining Like Terms

Now that we have expanded the expression using the distributive property, we can combine like terms to simplify it further. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with the variable $p$ raised to the power of 1.

$6pq + 24p = 6pq + 24p$

Since the terms $6pq$ and $24p$ have the same variable raised to the same power, we can combine them by adding their coefficients.

$6pq + 24p = (6 + 24)pq$

Simplifying the Expression

Now that we have combined like terms, we can simplify the expression further by evaluating the expression inside the parentheses.

$(6 + 24)pq = 30pq$

Therefore, the simplified expression is $30pq$.

Conclusion

In this article, we have explored how to use the distributive property and combine like terms to simplify the expression $2p(3q + 12)$. By following these steps, you will be able to simplify even the most complex algebraic expressions. Remember to always apply the distributive property and combine like terms to simplify expressions.

Tips and Tricks

• Always apply the distributive property when expanding expressions.
• Combine like terms to simplify expressions.
• Evaluate expressions inside parentheses to simplify them further.

Common Mistakes

• Failing to apply the distributive property when expanding expressions.
• Not combining like terms to simplify expressions.
• Evaluating expressions inside parentheses incorrectly.

Real-World Applications

The distributive property and combining like terms are essential skills in algebra that have real-world applications in fields such as physics, engineering, and economics. By mastering these skills, you will be able to solve complex problems and make informed decisions.

Practice Problems

1. Simplify the expression $3x(2y + 5)$ using the distributive property and combining like terms.
2. Simplify the expression $4p(2q - 3)$ using the distributive property and combining like terms.
3. Simplify the expression $2x(3y + 2)$ using the distributive property and combining like terms.

Answer Key

1. $6xy + 15x$
2. $8pq - 12p$
3. $6xy + 4x$

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 term outside the parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, simply multiply each term inside the parentheses by the term outside the parentheses. For example, in the expression $2p(3q + 12)$, we would multiply $2p$ by $3q$ and $2p$ by $12$.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, in the expression $6pq + 24p$, the terms $6pq$ and $24p$ are like terms because they both have the variable $p$ raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, simply add their coefficients. For example, in the expression $6pq + 24p$, we would add the coefficients $6$ and $24$ to get $30p$.

Q: What is the difference between the distributive property and combining like terms?

A: The distributive property is used to expand expressions by multiplying each term inside the parentheses by the term outside the parentheses. Combining like terms is used to simplify expressions by adding the coefficients of like terms.

Q: Can I simplify expressions without using the distributive property?

A: No, the distributive property is a necessary step in simplifying expressions. Without it, you would not be able to expand the expression and combine like terms.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Failing to apply the distributive property when expanding expressions
• Not combining like terms to simplify expressions
• Evaluating expressions inside parentheses incorrectly

Q: How do I know when to use the distributive property and when to combine like terms?

A: You should use the distributive property when expanding expressions and combining like terms when simplifying expressions. If you are unsure, try applying the distributive property and then combining like terms to see if you can simplify the expression.

Q: Can I use the distributive property and combining like terms to simplify expressions with variables raised to different powers?

A: Yes, you can use the distributive property and combining like terms to simplify expressions with variables raised to different powers. For example, in the expression $2x^2(3y + 4)$, you would apply the distributive property and then combine like terms to simplify the expression.

Q: Are there any other techniques I can use to simplify expressions?

A: Yes, there are several other techniques you can use to simplify expressions, including:

• Factoring expressions
• Canceling out common factors
• Using the order of operations to simplify expressions

Q: Can I use the distributive property and combining like terms to simplify expressions with fractions?

A: Yes, you can use the distributive property and combining like terms to simplify expressions with fractions. For example, in the expression $\frac{2x}{3}(4y + 5)$, you would apply the distributive property and then combine like terms to simplify the expression.

Q: Are there any online resources or tools that can help me practice simplifying expressions?

A: Yes, there are several online resources and tools that can help you practice simplifying expressions, including:

• Online algebra calculators
• Math websites and apps
• Practice problems and worksheets

By following the steps outlined in this article and practicing with online resources and tools, you will be able to simplify even the most complex algebraic expressions using the distributive property and combining like terms.