Which Of The Following Is The Product Of The Rational Expressions Shown Below? 7 X X + 3 ⋅ X + 1 X − 3 \frac{7x}{x+3} \cdot \frac{x+1}{x-3} X + 3 7 X ​ ⋅ X − 3 X + 1 ​ A. 7 X 2 + X X 2 − 6 \frac{7x^2+x}{x^2-6} X 2 − 6 7 X 2 + X ​ B. 7 X 2 + X X 2 − 9 \frac{7x^2+x}{x^2-9} X 2 − 9 7 X 2 + X ​ C. 7 X 2 + 7 X X 2 − 9 \frac{7x^2+7x}{x^2-9} X 2 − 9 7 X 2 + 7 X ​ D.

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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, using the given example of 7xx+3⋅x+1x−3\frac{7x}{x+3} \cdot \frac{x+1}{x-3} to illustrate the steps involved.

What are Rational Expressions?

A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be added, subtracted, multiplied, and divided, just like regular fractions. However, when multiplying rational expressions, we need to follow specific rules to simplify the resulting expression.

Multiplying Rational Expressions: The Rules

When multiplying rational expressions, we follow these rules:

  1. Multiply the numerators (the numbers on top) together.
  2. Multiply the denominators (the numbers on the bottom) together.
  3. Simplify the resulting expression by canceling out any common factors in the numerator and denominator.

Step-by-Step Solution

Now, let's apply these rules to the given example:

7xx+3⋅x+1x−3\frac{7x}{x+3} \cdot \frac{x+1}{x-3}

Step 1: Multiply the Numerators

Multiply the numerators together:

7x⋅(x+1)=7x2+7x7x \cdot (x+1) = 7x^2 + 7x

Step 2: Multiply the Denominators

Multiply the denominators together:

(x+3)⋅(x−3)=x2−9(x+3) \cdot (x-3) = x^2 - 9

Step 3: Simplify the Resulting Expression

Now, we have:

7x2+7xx2−9\frac{7x^2 + 7x}{x^2 - 9}

This expression cannot be simplified further, as there are no common factors in the numerator and denominator.

Conclusion

In conclusion, multiplying rational expressions involves following specific rules to simplify the resulting expression. By multiplying the numerators and denominators together and simplifying the resulting expression, we can find the product of the given rational expressions.

Answer

The correct answer is:

7x2+7xx2−9\boxed{\frac{7x^2 + 7x}{x^2 - 9}}

This matches option C, which is the correct answer.

Discussion

Multiplying rational expressions is a fundamental operation in algebra, and it's essential to understand the rules involved. By following these rules, we can simplify complex expressions and solve equations and inequalities.

Common Mistakes

When multiplying rational expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Not multiplying the numerators and denominators together correctly.
  • Not simplifying the resulting expression by canceling out common factors.
  • Not checking for any common factors in the numerator and denominator.

Tips and Tricks

Here are some tips and tricks to help you multiply rational expressions:

  • Always multiply the numerators and denominators together correctly.
  • Simplify the resulting expression by canceling out common factors.
  • Check for any common factors in the numerator and denominator.
  • Use a calculator or a computer algebra system to check your work.

Real-World Applications

Multiplying rational expressions has many real-world applications, including:

  • Solving equations and inequalities in physics and engineering.
  • Modeling population growth and decay in biology.
  • Analyzing financial data in economics.

Conclusion

Introduction

Multiplying rational expressions is a fundamental operation in algebra, and it's essential to understand the rules involved. In this article, we will answer some common questions about multiplying rational expressions, using the given example of 7xx+3⋅x+1x−3\frac{7x}{x+3} \cdot \frac{x+1}{x-3} to illustrate the steps involved.

Q: What is the product of the rational expressions 7xx+3⋅x+1x−3\frac{7x}{x+3} \cdot \frac{x+1}{x-3}?

A: The product of the rational expressions is 7x2+7xx2−9\frac{7x^2 + 7x}{x^2 - 9}.

Q: How do I multiply rational expressions?

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

  1. Multiply the numerators (the numbers on top) together.
  2. Multiply the denominators (the numbers on the bottom) together.
  3. Simplify the resulting expression by canceling out any common factors in the numerator and denominator.

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

A: When multiplying rational expressions, you multiply the numerators and denominators together, whereas when adding rational expressions, you add the numerators together and keep the same denominator.

Q: Can I simplify a rational expression by canceling out common factors in the numerator and denominator?

A: Yes, you can simplify a rational expression by canceling out common factors in the numerator and denominator. This is an essential step in simplifying rational expressions.

Q: How do I know if a rational expression can be simplified?

A: A rational expression can be simplified if there are common factors in the numerator and denominator. To simplify a rational expression, you need to factor the numerator and denominator and cancel out any common factors.

Q: What is the importance of multiplying rational expressions in real-world applications?

A: Multiplying rational expressions has many real-world applications, including solving equations and inequalities in physics and engineering, modeling population growth and decay in biology, and analyzing financial data in economics.

Q: Can I use a calculator or computer algebra system to multiply rational expressions?

A: Yes, you can use a calculator or computer algebra system to multiply rational expressions. However, it's essential to understand the rules involved and be able to simplify the resulting expression by canceling out common factors.

Q: What are some common mistakes to avoid when multiplying rational expressions?

A: Some common mistakes to avoid when multiplying rational expressions include:

  • Not multiplying the numerators and denominators together correctly.
  • Not simplifying the resulting expression by canceling out common factors.
  • Not checking for any common factors in the numerator and denominator.

Q: How can I practice multiplying rational expressions?

A: You can practice multiplying rational expressions by working through examples and exercises in your textbook or online resources. You can also use a calculator or computer algebra system to check your work and identify any mistakes.

Conclusion

In conclusion, multiplying rational expressions is a fundamental operation in algebra, and it's essential to understand the rules involved. By following these rules, we can simplify complex expressions and solve equations and inequalities. We hope this Q&A guide has been helpful in answering your questions about multiplying rational expressions.