Distribute The Multiplication Across The Addition Problem:$6p^2 \cdot (12p + 22y^4$\]

Introduction

In mathematics, the distributive property is a fundamental concept that allows us to simplify complex expressions by distributing the multiplication across the addition or subtraction. In this article, we will focus on distributing the multiplication across the addition problem, specifically the expression $6p^2 \cdot (12p + 22y^4)$. We will break down the problem step by step, providing a clear and concise explanation of each step.

Understanding the Distributive Property

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 means that we can distribute the multiplication across the addition or subtraction, resulting in the product of each term being multiplied by the other term.

Applying the Distributive Property to the Given Expression

Now that we have a clear understanding of the distributive property, let's apply it to the given expression $6p^2 \cdot (12p + 22y^4)$. To do this, we need to multiply each term inside the parentheses by the term outside the parentheses.

Step 1: Multiply the First Term

The first term inside the parentheses is $12p$. We need to multiply this term by $6p^2$.

$6p^2 \cdot 12p = 72p^3$

Step 2: Multiply the Second Term

The second term inside the parentheses is $22y^4$. We need to multiply this term by $6p^2$.

$6p^2 \cdot 22y^4 = 132p^2y^4$

Step 3: Combine the Results

Now that we have multiplied each term inside the parentheses by the term outside the parentheses, we can combine the results.

$6p^2 \cdot (12p + 22y^4) = 72p^3 + 132p^2y^4$

Conclusion

In this article, we have applied the distributive property to the expression $6p^2 \cdot (12p + 22y^4)$. We have broken down the problem step by step, providing a clear and concise explanation of each step. By distributing the multiplication across the addition, we have simplified the expression and obtained the final result.

Tips and Tricks

• When applying the distributive property, make sure to multiply each term inside the parentheses by the term outside the parentheses.
• Use the distributive property to simplify complex expressions and make them easier to work with.
• Practice, practice, practice! The more you practice applying the distributive property, the more comfortable you will become with it.

Common Mistakes to Avoid

• Failing to distribute the multiplication across the addition or subtraction.
• Not multiplying each term inside the parentheses by the term outside the parentheses.
• Not combining the results correctly.

Real-World Applications

The distributive property has many real-world applications, including:

• Simplifying complex expressions in algebra and calculus.
• Solving systems of linear equations.
• Working with polynomials and rational expressions.

Conclusion

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Q&A: Distributing the Multiplication Across the Addition Problem

Frequently Asked Questions

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 means that we can distribute the multiplication across the addition or subtraction, resulting in the product of each term being multiplied by the other term.

Q: How do I apply the distributive property to a problem?

A: To apply the distributive property, you need to multiply each term inside the parentheses by the term outside the parentheses. For example, if you have the expression $6p^2 \cdot (12p + 22y^4)$, you would multiply each term inside the parentheses by $6p^2$.

Q: What are some common mistakes to avoid when applying the distributive property?

A: Some common mistakes to avoid when applying the distributive property include:

• Failing to distribute the multiplication across the addition or subtraction.
• Not multiplying each term inside the parentheses by the term outside the parentheses.
• Not combining the results correctly.

Q: How do I simplify complex expressions using the distributive property?

A: To simplify complex expressions using the distributive property, you need to apply the property to each term inside the parentheses. For example, if you have the expression $(2x + 3)(x + 4)$, you would multiply each term inside the parentheses by the term outside the parentheses.

Q: What are some real-world applications of the distributive property?

A: The distributive property has many real-world applications, including:

• Simplifying complex expressions in algebra and calculus.
• Solving systems of linear equations.
• Working with polynomials and rational expressions.

Q: Can I use the distributive property to simplify expressions with more than two terms?

A: Yes, you can use the distributive property to simplify expressions with more than two terms. For example, if you have the expression $(2x + 3 + 4)(x + 4)$, you would multiply each term inside the parentheses by the term outside the parentheses.

Q: How do I know when to use the distributive property?

A: You should use the distributive property when you have an expression with parentheses and you need to simplify it. The distributive property is a powerful tool for simplifying complex expressions and making them easier to work with.

Additional Resources

• Khan Academy: Distributive Property
• Mathway: Distributive Property
• Wolfram Alpha: Distributive Property

Conclusion

In conclusion, the distributive property is a fundamental concept in mathematics that allows us to simplify complex expressions by distributing the multiplication across the addition or subtraction. By applying the distributive property to the expression $6p^2 \cdot (12p + 22y^4)$, we have obtained the final result of $72p^3 + 132p^2y^4$. We hope that this article has provided a clear and concise explanation of the distributive property and its applications.