Simplify The Expression:$\[ -7x^5 \cdot (-3x^3) \\]Calculate The Result:$\[ -7x - 3 = 21 \\]

Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements. When we multiply two or more terms, we need to apply the rules of exponents and signs to simplify the resulting expression. In this article, we will focus on simplifying the expression $-7x^5 \cdot (-3x^3)$ and discuss the steps involved in multiplying negative terms and variables.

Understanding the Rules of Exponents

Before we dive into the simplification process, let's review the rules of exponents. When we multiply two or more variables with the same base, we add their exponents. For example, $x^2 \cdot x^3 = x^{2+3} = x^5$. However, when we multiply variables with different bases, we keep their exponents separate. For instance, $x^2 \cdot y^3 = x^2 \cdot y^3$.

Simplifying the Expression

Now that we have reviewed the rules of exponents, let's simplify the expression $-7x^5 \cdot (-3x^3)$. To do this, we need to apply the rules of multiplication and exponents.

When we multiply two negative numbers, the result is a positive number. Therefore, $-7 \cdot (-3) = 21$. Now, let's focus on the variables. We have $x^5$ and $x^3$, which are both variables with the same base ($x$). According to the rules of exponents, we add their exponents: $x^5 \cdot x^3 = x^{5+3} = x^8$.

Combining the Results

Now that we have simplified the variables and the coefficients, let's combine the results. We have $21x^8$, which is the simplified expression.

Discussion

In this article, we have simplified the expression $-7x^5 \cdot (-3x^3)$ by applying the rules of exponents and signs. We have shown that when we multiply two negative numbers, the result is a positive number. We have also demonstrated how to add exponents when multiplying variables with the same base.

Conclusion

Simplifying expressions is an essential skill in algebra that helps us solve equations and manipulate mathematical statements. By applying the rules of exponents and signs, we can simplify complex expressions and arrive at a more manageable form. In this article, we have demonstrated how to simplify the expression $-7x^5 \cdot (-3x^3)$ and discussed the steps involved in multiplying negative terms and variables.

Example Problems

Here are some example problems that illustrate the concept of simplifying expressions:

• $-2x^3 \cdot (-5x^2) = ?$
• $3x^4 \cdot 2x^5 = ?$
• $-4x^2 \cdot (-3x^3) = ?$

Answer Key

Here are the answers to the example problems:

• $-2x^3 \cdot (-5x^2) = 10x^5$
• $3x^4 \cdot 2x^5 = 6x^9$
• $-4x^2 \cdot (-3x^3) = 12x^5$

Practice Problems

Here are some practice problems that will help you reinforce your understanding of simplifying expressions:

• Simplify the expression $-6x^4 \cdot (-2x^3)$.
• Simplify the expression $3x^2 \cdot 4x^5$.
• Simplify the expression $-5x^3 \cdot (-2x^2)$.

Answer Key

Here are the answers to the practice problems:

• $-6x^4 \cdot (-2x^3) = 12x^7$
• $3x^2 \cdot 4x^5 = 12x^7$
• $-5x^3 \cdot (-2x^2) = 10x^5$

Conclusion

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Introduction

In our previous article, we discussed how to simplify the expression $-7x^5 \cdot (-3x^3)$ by applying the rules of exponents and signs. We have shown that when we multiply two negative numbers, the result is a positive number. We have also demonstrated how to add exponents when multiplying variables with the same base.

In this article, we will provide a Q&A section to help you reinforce your understanding of simplifying expressions. We will answer some common questions and provide examples to illustrate the concept.

Q: What is the rule for multiplying negative numbers?

A: When we multiply two negative numbers, the result is a positive number. For example, $-7 \cdot (-3) = 21$.

Q: How do we add exponents when multiplying variables with the same base?

A: When we multiply variables with the same base, we add their exponents. For example, $x^2 \cdot x^3 = x^{2+3} = x^5$.

Q: What is the difference between multiplying variables with the same base and multiplying variables with different bases?

A: When we multiply variables with the same base, we add their exponents. For example, $x^2 \cdot x^3 = x^{2+3} = x^5$. However, when we multiply variables with different bases, we keep their exponents separate. For instance, $x^2 \cdot y^3 = x^2 \cdot y^3$.

Q: How do we simplify the expression $-2x^3 \cdot (-5x^2)$?

A: To simplify the expression $-2x^3 \cdot (-5x^2)$, we need to apply the rules of exponents and signs. We have $-2 \cdot (-5) = 10$, and $x^3 \cdot x^2 = x^{3+2} = x^5$. Therefore, the simplified expression is $10x^5$.

Q: How do we simplify the expression $3x^4 \cdot 2x^5$?

A: To simplify the expression $3x^4 \cdot 2x^5$, we need to apply the rules of exponents and signs. We have $3 \cdot 2 = 6$, and $x^4 \cdot x^5 = x^{4+5} = x^9$. Therefore, the simplified expression is $6x^9$.

Q: How do we simplify the expression $-4x^2 \cdot (-3x^3)$?

A: To simplify the expression $-4x^2 \cdot (-3x^3)$, we need to apply the rules of exponents and signs. We have $-4 \cdot (-3) = 12$, and $x^2 \cdot x^3 = x^{2+3} = x^5$. Therefore, the simplified expression is $12x^5$.

Conclusion

Simplifying expressions is a crucial skill in algebra that helps us solve equations and manipulate mathematical statements. By applying the rules of exponents and signs, we can simplify complex expressions and arrive at a more manageable form. In this article, we have provided a Q&A section to help you reinforce your understanding of simplifying expressions. We have answered some common questions and provided examples to illustrate the concept.

Practice Problems

Here are some practice problems that will help you reinforce your understanding of simplifying expressions:

• Simplify the expression $-6x^4 \cdot (-2x^3)$.
• Simplify the expression $3x^2 \cdot 4x^5$.
• Simplify the expression $-5x^3 \cdot (-2x^2)$.

Answer Key

Here are the answers to the practice problems:

• $-6x^4 \cdot (-2x^3) = 12x^7$
• $3x^2 \cdot 4x^5 = 12x^7$
• $-5x^3 \cdot (-2x^2) = 10x^5$

Conclusion

Simplifying expressions is a crucial skill in algebra that helps us solve equations and manipulate mathematical statements. By applying the rules of exponents and signs, we can simplify complex expressions and arrive at a more manageable form. In this article, we have provided a Q&A section to help you reinforce your understanding of simplifying expressions. We have answered some common questions and provided examples to illustrate the concept.