What Is The Product? \left(3 A^2 B^4\right)\left(-8 A B^3\right ]A. − 24 A B -24 A B − 24 Ab B. − 24 A 2 B 7 -24 A^2 B^7 − 24 A 2 B 7 C. − 24 A 2 B 12 -24 A^2 B^{12} − 24 A 2 B 12 D. − 24 A 3 B 7 -24 A^3 B^7 − 24 A 3 B 7

Understanding the Concept of Multiplying Algebraic Expressions

In mathematics, the product of two algebraic expressions is a fundamental concept that is used to simplify complex expressions and solve equations. When we multiply two algebraic expressions, we are essentially combining their variables and coefficients to form a new expression. In this article, we will explore the concept of multiplying algebraic expressions and provide a step-by-step guide on how to do it.

The Product of Two Algebraic Expressions

The product of two algebraic expressions is obtained by multiplying each term of the first expression by each term of the second expression. This means that we need to multiply the variables and coefficients of each term in both expressions and then combine the results.

For example, let's consider the product of two algebraic expressions: $\left(3 a^2 b^4\right)\left(-8 a b^3\right)$. To find the product, we need to multiply each term of the first expression by each term of the second expression.

Step-by-Step Guide to Multiplying Algebraic Expressions

To multiply two algebraic expressions, we need to follow these steps:

1. Identify the variables and coefficients: Identify the variables and coefficients in both expressions.
2. Multiply the variables: Multiply the variables of each term in both expressions.
3. Multiply the coefficients: Multiply the coefficients of each term in both expressions.
4. Combine the results: Combine the results of the multiplication of the variables and coefficients.

Multiplying the Variables and Coefficients

Let's apply these steps to the product of the two algebraic expressions: $\left(3 a^2 b^4\right)\left(-8 a b^3\right)$.

1. Identify the variables and coefficients: The variables are $a$ and $b$, and the coefficients are $3$ and $-8$.
2. Multiply the variables: Multiply the variables of each term in both expressions:
   • $a^2 \cdot a = a^3$
   • $b^4 \cdot b^3 = b^7$
3. Multiply the coefficients: Multiply the coefficients of each term in both expressions:
   • $3 \cdot -8 = -24$
4. Combine the results: Combine the results of the multiplication of the variables and coefficients:
   • $-24 a^3 b^7$

Conclusion

In conclusion, the product of two algebraic expressions is obtained by multiplying each term of the first expression by each term of the second expression. By following the steps outlined in this article, we can simplify complex expressions and solve equations. The product of the two algebraic expressions $\left(3 a^2 b^4\right)\left(-8 a b^3\right)$ is $-24 a^3 b^7$.

Frequently Asked Questions

• What is the product of two algebraic expressions? The product of two algebraic expressions is obtained by multiplying each term of the first expression by each term of the second expression.
• How do I multiply algebraic expressions? To multiply algebraic expressions, you need to follow these steps: identify the variables and coefficients, multiply the variables, multiply the coefficients, and combine the results.
• What is the product of $\left(3 a^2 b^4\right)\left(-8 a b^3\right)$? The product of $\left(3 a^2 b^4\right)\left(-8 a b^3\right)$ is $-24 a^3 b^7$.

Final Answer

The final answer is $\boxed{-24 a^3 b^7}$.

Understanding the Concept of Multiplying Algebraic Expressions

Multiplying algebraic expressions is a fundamental concept in mathematics that is used to simplify complex expressions and solve equations. In this article, we will provide answers to frequently asked questions about multiplying algebraic expressions.

Q&A: Multiplying Algebraic Expressions

Q: What is the product of two algebraic expressions?

A: The product of two algebraic expressions is obtained by multiplying each term of the first expression by each term of the second expression.

Q: How do I multiply algebraic expressions?

A: To multiply algebraic expressions, you need to follow these steps:

1. Identify the variables and coefficients: Identify the variables and coefficients in both expressions.
2. Multiply the variables: Multiply the variables of each term in both expressions.
3. Multiply the coefficients: Multiply the coefficients of each term in both expressions.
4. Combine the results: Combine the results of the multiplication of the variables and coefficients.

Q: What is the product of $\left(3 a^2 b^4\right)\left(-8 a b^3\right)$?

A: The product of $\left(3 a^2 b^4\right)\left(-8 a b^3\right)$ is $-24 a^3 b^7$.

Q: How do I handle negative coefficients when multiplying algebraic expressions?

A: When multiplying algebraic expressions with negative coefficients, you need to multiply the coefficients as you would with positive coefficients, but then apply the rule that a negative times a negative is a positive.

Q: Can I multiply algebraic expressions with different variables?

A: Yes, you can multiply algebraic expressions with different variables. When multiplying variables with different bases, you need to multiply the exponents of the variables.

Q: How do I simplify the product of two algebraic expressions?

A: To simplify the product of two algebraic expressions, you need to combine like terms and apply the rules of exponents.

Q: What is the difference between multiplying algebraic expressions and adding or subtracting them?

A: Multiplying algebraic expressions involves combining the variables and coefficients of each term in both expressions, whereas adding or subtracting algebraic expressions involves combining like terms.

Q: Can I use a calculator to multiply algebraic expressions?

A: Yes, you can use a calculator to multiply algebraic expressions, but it's always a good idea to double-check your work to ensure that you have the correct answer.

Conclusion

In conclusion, multiplying algebraic expressions is a fundamental concept in mathematics that is used to simplify complex expressions and solve equations. By following the steps outlined in this article, you can multiply algebraic expressions with confidence. Remember to identify the variables and coefficients, multiply the variables, multiply the coefficients, and combine the results to get the product of two algebraic expressions.

Final Tips

• Practice, practice, practice: The more you practice multiplying algebraic expressions, the more comfortable you will become with the process.
• Use a calculator: If you're struggling to multiply algebraic expressions, try using a calculator to check your work.
• Double-check your work: Always double-check your work to ensure that you have the correct answer.

Final Answer

The final answer is $\boxed{-24 a^3 b^7}$.