Find The Product:b. $\frac{(x+1)}{8} \cdot \frac{9x}{(x+3)}$

Feb 28, 2025 by ADMIN 61 views

**Finding the Product of Two Rational Expressions: A Step-by-Step Guide**

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Introduction

In algebra, rational expressions are a fundamental concept that plays a crucial role in solving various mathematical problems. A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. When we are given two rational expressions, we often need to find their product. In this article, we will explore how to find the product of two rational expressions, using the given problem as an example.

Understanding the Problem

The problem asks us to find the product of two rational expressions:

\frac{(x+1)}{8} \cdot \frac{9x}{(x+3)}

To find the product, we need to multiply the numerators and denominators separately.

Multiplying Numerators and Denominators

When multiplying two rational expressions, we multiply the numerators together and the denominators together. In this case, we have:

\frac{(x+1)}{8} \cdot \frac{9x}{(x+3)} = \frac{(x+1) \cdot 9x}{8 \cdot (x+3)}

Simplifying the Expression

Now that we have multiplied the numerators and denominators, we can simplify the expression by canceling out any common factors. In this case, we can cancel out the common factor of $x$ in the numerator and denominator:

\frac{(x+1) \cdot 9x}{8 \cdot (x+3)} = \frac{9x(x+1)}{8(x+3)}

Final Answer

The final answer is:

\frac{9x(x+1)}{8(x+3)}

Why is Simplifying Important?

Simplifying rational expressions is an essential step in solving mathematical problems. When we simplify an expression, we are reducing it to its simplest form, which makes it easier to work with. In this case, simplifying the expression allowed us to cancel out the common factor of $x$ , making the expression easier to read and understand.

Real-World Applications

Rational expressions have numerous real-world applications in fields such as engineering, economics, and physics. For example, in engineering, rational expressions are used to model the behavior of electrical circuits and mechanical systems. In economics, rational expressions are used to model the behavior of markets and economies. In physics, rational expressions are used to model the behavior of physical systems, such as the motion of objects and the behavior of waves.

Conclusion

In conclusion, finding the product of two rational expressions is a fundamental concept in algebra that has numerous real-world applications. By following the steps outlined in this article, we can simplify rational expressions and find their product. Remember, simplifying rational expressions is an essential step in solving mathematical problems, and it can make a big difference in the accuracy and efficiency of our solutions.

Tips and Tricks

Here are some tips and tricks to help you find the product of two rational expressions:

Read the problem carefully: Before starting to solve the problem, read it carefully and make sure you understand what is being asked.
Identify the numerators and denominators: Identify the numerators and denominators of each rational expression and multiply them separately.
Simplify the expression: Simplify the expression by canceling out any common factors.
Check your answer: Check your answer to make sure it is correct.

Common Mistakes to Avoid

Here are some common mistakes to avoid when finding the product of two rational expressions:

Not simplifying the expression: Failing to simplify the expression can lead to incorrect answers.
Not canceling out common factors: Failing to cancel out common factors can lead to incorrect answers.
Not checking the answer: Failing to check the answer can lead to incorrect answers.

Conclusion

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Q: What is the product of two rational expressions?

A: The product of two rational expressions is the result of multiplying the two expressions together. It is a new rational expression that is the product of the two original expressions.

Q: How do I find the product of two rational expressions?

A: To find the product of two rational expressions, you need to multiply the numerators together and the denominators together. Then, simplify the resulting expression by canceling out any common factors.

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

A: When multiplying rational expressions, you multiply the numerators together and the denominators together. When adding or subtracting rational expressions, you need to find a common denominator and then add or subtract the numerators.

Q: Can I simplify a rational expression before multiplying it with another rational expression?

A: Yes, you can simplify a rational expression before multiplying it with another rational expression. In fact, simplifying the expression before multiplying it can make the multiplication process easier and more efficient.

Q: What is the importance of simplifying rational expressions?

A: Simplifying rational expressions is important because it helps to reduce the complexity of the expression and makes it easier to work with. Simplifying an expression can also help to identify any common factors that can be canceled out, which can make the expression easier to read and understand.

Q: Can I use a calculator to find the product of two rational expressions?

A: Yes, you can use a calculator to find the product of two rational expressions. However, it's always a good idea to simplify the expression by hand before using a calculator to check your answer.

Q: What are some common mistakes to avoid when finding the product of two rational expressions?

A: Some common mistakes to avoid when finding the product of two rational expressions include:

Not simplifying the expression before multiplying it
Not canceling out common factors
Not checking the answer
Not using the correct order of operations

Q: Can I use the product of two rational expressions to solve real-world problems?

A: Yes, you can use the product of two rational expressions to solve real-world problems. Rational expressions are used in a variety of fields, including engineering, economics, and physics, to model the behavior of systems and make predictions about future outcomes.

Q: How do I know if I have found the correct product of two rational expressions?

A: To know if you have found the correct product of two rational expressions, you need to check your answer by plugging it back into the original problem and making sure it is true. You can also use a calculator to check your answer.

Q: Can I use the product of two rational expressions to find the sum or difference of two rational expressions?

A: No, you cannot use the product of two rational expressions to find the sum or difference of two rational expressions. To find the sum or difference of two rational expressions, you need to add or subtract the numerators and keep the denominator the same.

Q: What is the final answer to the problem of finding the product of two rational expressions?

A: The final answer to the problem of finding the product of two rational expressions is the simplified expression that results from multiplying the two original expressions together.