What Is The Product? \left(7 X^2 Y^3\right)\left(3 X^5 Y^8\right ]A. 10 X 7 Y 11 10 X^7 Y^{11} 10 X 7 Y 11 B. 10 X 10 Y 24 10 X^{10} Y^{24} 10 X 10 Y 24 C. 21 X 7 Y 11 21 X^7 Y^{11} 21 X 7 Y 11 D. 21 X 10 Y 24 21 X^{10} Y^{24} 21 X 10 Y 24

Understanding the Concept of Multiplication in Algebra

In algebra, multiplication is a fundamental operation that allows us to combine two or more expressions to form a new expression. When we multiply two expressions, we are essentially combining their terms to create a new expression. In this article, we will explore the concept of multiplication in algebra and learn how to find the product of two algebraic expressions.

The Product of Two Algebraic Expressions

The product of two algebraic expressions is a new expression that is formed by multiplying the two expressions together. When we multiply two expressions, we multiply each term in the first expression by each term in the second expression. The resulting expression is a new expression that contains all the terms from both expressions.

Example: Multiplying Two Algebraic Expressions

Let's consider an example to illustrate the concept of multiplication in algebra. Suppose we want to find the product of the two expressions $\left(7 x^2 y^3\right)$ and $\left(3 x^5 y^8\right)$. To find the product, we multiply each term in the first expression by each term in the second expression.

$\left(7 x^2 y^3\right)\left(3 x^5 y^8\right)$

To multiply these expressions, we multiply the coefficients (the numbers in front of the variables) and add the exponents of the variables.

Multiplying Coefficients

The coefficients of the two expressions are 7 and 3. To multiply these coefficients, we simply multiply them together.

$7 \times 3 = 21$

Adding Exponents

The variables in the two expressions are $x$ and $y$. To add the exponents of these variables, we add the exponents of $x$ and $y$ separately.

$x^2 \times x^5 = x^{2+5} = x^7$

$y^3 \times y^8 = y^{3+8} = y^{11}$

The Product of the Two Expressions

Now that we have multiplied the coefficients and added the exponents, we can write the product of the two expressions.

$\left(7 x^2 y^3\right)\left(3 x^5 y^8\right) = 21 x^7 y^{11}$

Conclusion

In this article, we learned how to find the product of two algebraic expressions. We saw that multiplication in algebra involves multiplying the coefficients and adding the exponents of the variables. By following these steps, we can find the product of two expressions and simplify complex algebraic expressions.

Key Takeaways

• Multiplication in algebra involves multiplying the coefficients and adding the exponents of the variables.
• To multiply two expressions, we multiply each term in the first expression by each term in the second expression.
• The product of two expressions is a new expression that contains all the terms from both expressions.

Practice Problems

1. Find the product of the two expressions $\left(4 x^3 y^2\right)$ and $\left(2 x^4 y^5\right)$.
2. Find the product of the two expressions $\left(3 x^2 y^3\right)$ and $\left(5 x^6 y^9\right)$.
3. Find the product of the two expressions $\left(2 x^4 y^7\right)$ and $\left(3 x^2 y^3\right)$.

Answer Key

1. $\left(4 x^3 y^2\right)\left(2 x^4 y^5\right) = 8 x^7 y^7$
2. $\left(3 x^2 y^3\right)\left(5 x^6 y^9\right) = 15 x^8 y^{12}$
3. $\left(2 x^4 y^7\right)\left(3 x^2 y^3\right) = 6 x^6 y^{10}$
   Frequently Asked Questions (FAQs) About Multiplication in Algebra ====================================================================

Q: What is the product of two algebraic expressions?

A: The product of two algebraic expressions is a new expression that is formed by multiplying the two expressions together. When we multiply two expressions, we multiply each term in the first expression by each term in the second expression.

Q: How do I multiply two algebraic expressions?

A: To multiply two algebraic expressions, you multiply each term in the first expression by each term in the second expression. You multiply the coefficients (the numbers in front of the variables) and add the exponents of the variables.

Q: What is the rule for multiplying variables with the same base?

A: When multiplying variables with the same base, you add the exponents. For example, $x^2 \times x^5 = x^{2+5} = x^7$.

Q: What is the rule for multiplying variables with different bases?

A: When multiplying variables with different bases, you keep the variables separate. For example, $x^2 \times y^3 = x^2y^3$.

Q: How do I multiply a coefficient by a variable?

A: To multiply a coefficient by a variable, you multiply the coefficient by the variable. For example, $2 \times x^3 = 2x^3$.

Q: How do I multiply two expressions with the same variable but different exponents?

A: To multiply two expressions with the same variable but different exponents, you add the exponents. For example, $x^2 \times x^5 = x^{2+5} = x^7$.

Q: Can I multiply two expressions with the same variable but different coefficients?

A: Yes, you can multiply two expressions with the same variable but different coefficients. You multiply the coefficients and keep the variable the same. For example, $2x \times 3x = 6x^2$.

Q: How do I multiply two expressions with different variables?

A: To multiply two expressions with different variables, you keep the variables separate. For example, $x^2 \times y^3 = x^2y^3$.

Q: Can I multiply two expressions with different variables and coefficients?

A: Yes, you can multiply two expressions with different variables and coefficients. You multiply the coefficients and keep the variables separate. For example, $2x \times 3y = 6xy$.

Q: What is the product of the two expressions $\left(4 x^3 y^2\right)$ and $\left(2 x^4 y^5\right)$?

A: The product of the two expressions $\left(4 x^3 y^2\right)$ and $\left(2 x^4 y^5\right)$ is $8 x^7 y^7$.

Q: What is the product of the two expressions $\left(3 x^2 y^3\right)$ and $\left(5 x^6 y^9\right)$?

A: The product of the two expressions $\left(3 x^2 y^3\right)$ and $\left(5 x^6 y^9\right)$ is $15 x^8 y^{12}$.

Q: What is the product of the two expressions $\left(2 x^4 y^7\right)$ and $\left(3 x^2 y^3\right)$?

A: The product of the two expressions $\left(2 x^4 y^7\right)$ and $\left(3 x^2 y^3\right)$ is $6 x^6 y^{10}$.

Conclusion

In this article, we have answered some frequently asked questions about multiplication in algebra. We have covered topics such as multiplying variables with the same base, multiplying variables with different bases, multiplying coefficients by variables, and multiplying expressions with the same variable but different exponents. We have also provided examples to illustrate these concepts.