Distribute The Multiplication Across The Subtraction Problem:$\[ 5 J^3 \cdot (22 J^5 - 13 P^4) \\]

Introduction

In algebra, when we encounter a problem that involves multiplying a term by a binomial or trinomial, we need to distribute the multiplication across the terms inside the parentheses. This process is also known as the distributive property of multiplication over addition. In this article, we will learn how to distribute the multiplication across the subtraction problem, using the given example: $5 j^3 \cdot (22 j^5 - 13 p^4)$.

Understanding the Distributive Property

The distributive property of multiplication over addition states that for any real numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

This property can be extended to include subtraction by rewriting it as an addition problem. For example, a(b - c) can be rewritten as a(b + (-c)), where (-c) is the opposite of c.

Applying the Distributive Property to the Given Problem

Now, let's apply the distributive property to the given problem: $5 j^3 \cdot (22 j^5 - 13 p^4)$. To do this, we need to multiply the term $5 j^3$ by each term inside the parentheses.

Step 1: Multiply $5 j^3$ by $22 j^5$

To multiply $5 j^3$ by $22 j^5$, we need to multiply the coefficients (5 and 22) and add the exponents of the variables (j). The result is:

$5 j^3 \cdot 22 j^5 = 110 j^{3+5} = 110 j^8$

Step 2: Multiply $5 j^3$ by $-13 p^4$

To multiply $5 j^3$ by $-13 p^4$, we need to multiply the coefficients (5 and -13) and add the exponents of the variables (j and p). The result is:

$5 j^3 \cdot (-13 p^4) = -65 j^3 p^4$

Step 3: Combine the Results

Now, we need to combine the results of the two multiplications. Since the problem involves subtraction, we need to subtract the second result from the first result:

$110 j^8 - 65 j^3 p^4$

Conclusion

In this article, we learned how to distribute the multiplication across the subtraction problem using the given example: $5 j^3 \cdot (22 j^5 - 13 p^4)$. We applied the distributive property of multiplication over addition to multiply the term $5 j^3$ by each term inside the parentheses and then combined the results. The final answer is $110 j^8 - 65 j^3 p^4$.

Example Problems

Here are some example problems that involve distributing the multiplication across the subtraction problem:

Example 1

$3 x^2 \cdot (4 x^3 - 2 y^2)$

Solution

$3 x^2 \cdot 4 x^3 = 12 x^{2+3} = 12 x^5$

$3 x^2 \cdot (-2 y^2) = -6 x^2 y^2$

$12 x^5 - 6 x^2 y^2$

Example 2

$2 y^3 \cdot (5 z^4 - 3 w^2)$

Solution

$2 y^3 \cdot 5 z^4 = 10 y^3 z^4$

$2 y^3 \cdot (-3 w^2) = -6 y^3 w^2$

$10 y^3 z^4 - 6 y^3 w^2$

Tips and Tricks

Here are some tips and tricks to help you distribute the multiplication across the subtraction problem:

• Make sure to multiply each term inside the parentheses by the term outside the parentheses.
• Use the distributive property of multiplication over addition to multiply the terms.
• Combine the results of the multiplications by adding or subtracting the terms.
• Simplify the expression by combining like terms.

Introduction

In our previous article, we learned how to distribute the multiplication across the subtraction problem using the given example: $5 j^3 \cdot (22 j^5 - 13 p^4)$. In this article, we will answer some frequently asked questions about distributing the multiplication across the subtraction problem.

Q&A

Q: What is the distributive property of multiplication over addition?

A: The distributive property of multiplication over addition states that for any real numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

This property can be extended to include subtraction by rewriting it as an addition problem. For example, a(b - c) can be rewritten as a(b + (-c)), where (-c) is the opposite of c.

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

A: To apply the distributive property to a problem, you need to multiply each term inside the parentheses by the term outside the parentheses. For example, if you have the problem $5 j^3 \cdot (22 j^5 - 13 p^4)$, you would multiply $5 j^3$ by $22 j^5$ and $5 j^3$ by $-13 p^4$.

Q: What is the difference between distributing the multiplication across the addition problem and the subtraction problem?

A: The main difference between distributing the multiplication across the addition problem and the subtraction problem is that the subtraction problem involves subtracting one term from another. When distributing the multiplication across the subtraction problem, you need to multiply each term inside the parentheses by the term outside the parentheses and then subtract the second result from the first result.

Q: How do I simplify the expression after distributing the multiplication?

A: After distributing the multiplication, you need to simplify the expression by combining like terms. This involves adding or subtracting terms that have the same variable and exponent.

Q: What are some common mistakes to avoid when distributing the multiplication?

A: Some common mistakes to avoid when distributing the multiplication include:

• Forgetting to multiply each term inside the parentheses by the term outside the parentheses.
• Not simplifying the expression after distributing the multiplication.
• Not combining like terms.

Q: How can I practice distributing the multiplication across the subtraction problem?

A: You can practice distributing the multiplication across the subtraction problem by working through example problems and exercises. You can also use online resources and practice tests to help you improve your skills.

Example Problems

Here are some example problems that involve distributing the multiplication across the subtraction problem:

Example 1

$3 x^2 \cdot (4 x^3 - 2 y^2)$

Solution

$3 x^2 \cdot 4 x^3 = 12 x^{2+3} = 12 x^5$

$3 x^2 \cdot (-2 y^2) = -6 x^2 y^2$

$12 x^5 - 6 x^2 y^2$

Example 2

$2 y^3 \cdot (5 z^4 - 3 w^2)$

Solution

$2 y^3 \cdot 5 z^4 = 10 y^3 z^4$

$2 y^3 \cdot (-3 w^2) = -6 y^3 w^2$

$10 y^3 z^4 - 6 y^3 w^2$

Tips and Tricks

Here are some tips and tricks to help you distribute the multiplication across the subtraction problem:

• Make sure to multiply each term inside the parentheses by the term outside the parentheses.
• Use the distributive property of multiplication over addition to multiply the terms.
• Combine the results of the multiplications by adding or subtracting the terms.
• Simplify the expression by combining like terms.
• Practice, practice, practice!

By following these tips and tricks, you can become proficient in distributing the multiplication across the subtraction problem and solve complex algebraic expressions with ease.