Which Of The Following Is The Product Of The Rational Expressions Shown Below?${ \frac{x+3}{x+2} \cdot \frac{x-3}{x-2} }$A. { \frac{9}{4}$}$B. { \frac{x 2-4}{x 2-9}$}$C. { \frac{x 2}{x 2-4}$}$D.

Introduction

Rational expressions are a fundamental concept in algebra, and multiplying them is a crucial operation in solving equations and inequalities. In this article, we will explore the process of multiplying rational expressions, with a focus on the given problem: $\frac{x+3}{x+2} \cdot \frac{x-3}{x-2}$. We will break down the solution step by step, using clear and concise language to ensure that readers understand the concept.

What are Rational Expressions?

Before we dive into the solution, let's briefly define what rational expressions are. A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified, added, subtracted, multiplied, and divided, just like regular fractions.

Multiplying Rational Expressions

To multiply rational expressions, we follow a simple rule: multiply the numerators together and multiply the denominators together. This is similar to multiplying regular fractions.

Step 1: Multiply the Numerators

The first step is to multiply the numerators together. In this case, we have $(x+3)$ and $(x-3)$, which we multiply together:

$(x+3) \cdot (x-3) = x^2 - 3x + 3x - 9 = x^2 - 9$

Step 2: Multiply the Denominators

Next, we multiply the denominators together. In this case, we have $(x+2)$ and $(x-2)$, which we multiply together:

$(x+2) \cdot (x-2) = x^2 - 4$

Step 3: Write the Product

Now that we have multiplied the numerators and denominators, we can write the product of the rational expressions:

$\frac{x^2 - 9}{x^2 - 4}$

Which of the Following is the Product?

Now that we have found the product of the rational expressions, we can compare it to the given options:

A. $\frac{9}{4}$ B. $\frac{x^2-4}{x^2-9}$ C. $\frac{x^2}{x^2-4}$ D. $\frac{x^2-9}{x^2-4}$

The correct answer is D. $\frac{x^2-9}{x^2-4}$.

Conclusion

Multiplying rational expressions is a straightforward process that involves multiplying the numerators together and multiplying the denominators together. By following these steps, we can simplify complex rational expressions and solve equations and inequalities. In this article, we have explored the process of multiplying rational expressions, with a focus on the given problem. We have also compared the product to the given options and found the correct answer.

Frequently Asked Questions

• Q: What is the product of the rational expressions $\frac{x+3}{x+2} \cdot \frac{x-3}{x-2}$? A: The product is $\frac{x^2-9}{x^2-4}$.
• Q: How do I multiply rational expressions? A: To multiply rational expressions, multiply the numerators together and multiply the denominators together.
• Q: What is the difference between multiplying rational expressions and multiplying regular fractions? A: The main difference is that rational expressions can contain variables and/or constants in the numerator and/or denominator.

Additional Resources

For more information on multiplying rational expressions, check out the following resources:

• Khan Academy: Multiplying Rational Expressions
• Mathway: Multiplying Rational Expressions
• Wolfram Alpha: Multiplying Rational Expressions

References

• "Algebra and Trigonometry" by Michael Sullivan
• "College Algebra" by James Stewart
• "Rational Expressions" by Math Open Reference
  Rational Expressions Q&A ==========================

Introduction

Rational expressions are a fundamental concept in algebra, and understanding how to work with them is crucial for solving equations and inequalities. In this article, we will answer some of the most frequently asked questions about rational expressions, with a focus on multiplication.

Q: What is the product of the rational expressions $\frac{x+3}{x+2} \cdot \frac{x-3}{x-2}$?

A: The product is $\frac{x^2-9}{x^2-4}$.

Q: How do I multiply rational expressions?

A: To multiply rational expressions, multiply the numerators together and multiply the denominators together. For example, to multiply $\frac{x+3}{x+2} \cdot \frac{x-3}{x-2}$, you would multiply the numerators together to get $x^2 - 9$ and multiply the denominators together to get $x^2 - 4$.

Q: What is the difference between multiplying rational expressions and multiplying regular fractions?

A: The main difference is that rational expressions can contain variables and/or constants in the numerator and/or denominator. When multiplying regular fractions, you can simply multiply the numerators together and multiply the denominators together. However, when multiplying rational expressions, you need to be careful to multiply the variables and constants correctly.

Q: Can I simplify rational expressions before multiplying them?

A: Yes, you can simplify rational expressions before multiplying them. In fact, simplifying rational expressions can make it easier to multiply them. For example, if you have the rational expression $\frac{x+3}{x+2}$, you can simplify it by canceling out the common factor of $x+2$ to get $\frac{1}{1}$.

Q: How do I handle negative exponents when multiplying rational expressions?

A: When multiplying rational expressions with negative exponents, you can use the rule that $a^{-n} = \frac{1}{a^n}$. For example, if you have the rational expression $\frac{x^{-2}}{x^{-1}}$, you can rewrite it as $\frac{1}{x^2} \cdot \frac{x}{1}$.

Q: Can I multiply rational expressions with different variables?

A: Yes, you can multiply rational expressions with different variables. For example, if you have the rational expressions $\frac{x+3}{x+2}$ and $\frac{y-3}{y-2}$, you can multiply them together to get $\frac{(x+3)(y-3)}{(x+2)(y-2)}$.

Q: How do I handle zero denominators when multiplying rational expressions?

A: When multiplying rational expressions, you need to be careful to avoid zero denominators. If the denominator of one of the rational expressions is zero, you cannot multiply it by another rational expression. For example, if you have the rational expression $\frac{x+3}{x+2}$ and the denominator is zero, you cannot multiply it by $\frac{x-3}{x-2}$.

Q: Can I divide rational expressions?

A: Yes, you can divide rational expressions. To divide rational expressions, you can multiply the numerator by the reciprocal of the denominator. For example, to divide $\frac{x+3}{x+2}$ by $\frac{x-3}{x-2}$, you can multiply the numerator by the reciprocal of the denominator to get $\frac{(x+3)(x-2)}{(x+2)(x-3)}$.

Conclusion

Rational expressions are a fundamental concept in algebra, and understanding how to work with them is crucial for solving equations and inequalities. In this article, we have answered some of the most frequently asked questions about rational expressions, with a focus on multiplication. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of rational expressions.

Frequently Asked Questions

• Q: What is the product of the rational expressions $\frac{x+3}{x+2} \cdot \frac{x-3}{x-2}$? A: The product is $\frac{x^2-9}{x^2-4}$.
• Q: How do I multiply rational expressions? A: To multiply rational expressions, multiply the numerators together and multiply the denominators together.
• Q: Can I simplify rational expressions before multiplying them? A: Yes, you can simplify rational expressions before multiplying them.

Additional Resources

For more information on rational expressions, check out the following resources:

• Khan Academy: Rational Expressions
• Mathway: Rational Expressions
• Wolfram Alpha: Rational Expressions

References

• "Algebra and Trigonometry" by Michael Sullivan
• "College Algebra" by James Stewart
• "Rational Expressions" by Math Open Reference