Distribute The Multiplication Across The Subtraction Problem.$6x^5 \cdot (23t - 3x^3$\]

Understanding the Problem

When dealing with algebraic expressions, it's essential to understand the order of operations and how to distribute multiplication across various components of an expression. In this article, we'll focus on distributing the multiplication across a subtraction problem, specifically the expression $6x^5 \cdot (23t - 3x^3)$.

The Order of Operations

Before we dive into the problem, let's review the order of operations, which is a set of rules that dictate the order in which we perform mathematical operations. The order of operations is often remembered using the acronym PEMDAS, which stands for:

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

Distributing Multiplication Across a Subtraction Problem

Now that we've reviewed the order of operations, let's apply it to the given expression: $6x^5 \cdot (23t - 3x^3)$. To distribute the multiplication across the subtraction problem, we need to multiply the term $6x^5$ by each component of the expression inside the parentheses.

Step 1: Multiply $6x^5$ by $23t$

To multiply $6x^5$ by $23t$, we need to multiply the coefficients (6 and 23) and the variables ($x^5$ and $t$). When multiplying variables with the same base, we add their exponents. Therefore, the product of $6x^5$ and $23t$ is:

$6x^5 \cdot 23t = 138x^5t$

Step 2: Multiply $6x^5$ by $-3x^3$

To multiply $6x^5$ by $-3x^3$, we need to multiply the coefficients (6 and -3) and the variables ($x^5$ and $x^3$). When multiplying variables with the same base, we add their exponents. Therefore, the product of $6x^5$ and $-3x^3$ is:

$6x^5 \cdot -3x^3 = -18x^8$

Step 3: Combine the Results

Now that we've multiplied $6x^5$ by each component of the expression inside the parentheses, we can combine the results:

$6x^5 \cdot (23t - 3x^3) = 138x^5t - 18x^8$

Conclusion

Distributing multiplication across a subtraction problem involves multiplying the term by each component of the expression inside the parentheses. By following the order of operations and applying the rules of exponentiation, we can simplify complex algebraic expressions. In this article, we've demonstrated how to distribute the multiplication across the subtraction problem $6x^5 \cdot (23t - 3x^3)$, resulting in the simplified expression $138x^5t - 18x^8$.

Example Problems

To reinforce your understanding of distributing multiplication across a subtraction problem, try the following example problems:

1. Distribute the multiplication across the subtraction problem: $4y^2 \cdot (9z - 2y)$
2. Distribute the multiplication across the subtraction problem: $3x^4 \cdot (2x - 5y)$

Practice Exercises

To practice distributing multiplication across a subtraction problem, try the following exercises:

1. Distribute the multiplication across the subtraction problem: $2a^3 \cdot (4b - 3a)$
2. Distribute the multiplication across the subtraction problem: $5c^2 \cdot (2c - 3d)$

Common Mistakes

When distributing multiplication across a subtraction problem, it's essential to avoid common mistakes, such as:

• Forgetting to multiply the coefficients by each component of the expression inside the parentheses.
• Failing to apply the rules of exponentiation when multiplying variables with the same base.
• Not combining the results correctly.

Frequently Asked Questions

In this article, we'll address some of the most common questions related to distributing multiplication across a subtraction problem.

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which we perform mathematical operations. The order of operations is often remembered using the acronym PEMDAS, which stands for:

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

Q: How do I distribute multiplication across a subtraction problem?

A: To distribute multiplication across a subtraction problem, you need to multiply the term by each component of the expression inside the parentheses. This involves multiplying the coefficients (numbers) and the variables (letters with exponents) separately.

Q: What happens when I multiply variables with the same base?

A: When you multiply variables with the same base, you add their exponents. For example, if you multiply $x^2$ by $x^3$, the result is $x^{2+3} = x^5$.

Q: Can I distribute multiplication across a subtraction problem with multiple terms?

A: Yes, you can distribute multiplication across a subtraction problem with multiple terms. For example, if you have the expression $2x^2 \cdot (3y - 4z + 5w)$, you would multiply $2x^2$ by each term inside the parentheses separately.

Q: What is the difference between distributing multiplication and factoring?

A: Distributing multiplication involves multiplying a term by each component of an expression, while factoring involves expressing an expression as a product of simpler expressions.

Q: Can I use the distributive property to simplify complex expressions?

A: Yes, the distributive property can be used to simplify complex expressions by distributing multiplication across various components of the expression.

Q: What are some common mistakes to avoid when distributing multiplication across a subtraction problem?

A: Some common mistakes to avoid when distributing multiplication across a subtraction problem include:

• Forgetting to multiply the coefficients by each component of the expression inside the parentheses.
• Failing to apply the rules of exponentiation when multiplying variables with the same base.
• Not combining the results correctly.

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

A: You can practice distributing multiplication across a subtraction problem by working through example problems and exercises. Try to distribute multiplication across various expressions, including those with multiple terms and variables with exponents.

Q: What are some real-world applications of distributing multiplication across a subtraction problem?

A: Distributing multiplication across a subtraction problem has many real-world applications, including:

• Simplifying complex algebraic expressions in physics and engineering.
• Modeling population growth and decline in biology.
• Analyzing financial data and making predictions in economics.

Conclusion

Distributing multiplication across a subtraction problem is a fundamental concept in algebra that has many real-world applications. By understanding the order of operations and the rules of exponentiation, you can simplify complex expressions and make predictions in various fields. Remember to practice distributing multiplication across a subtraction problem to build your skills and confidence.