Select The Correct Answer.Assuming No Denominator Equals Zero, Which Expression Is Equivalent To The Given Expression?Given Expression:$ \frac{x+7}{7x+35} \cdot \frac{x^2-3x-40}{x-8} }$Options A. { \frac{x+5 {7(x-8)}$}$ B.

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will explore how to simplify a given algebraic expression by factoring and canceling out common factors. We will also discuss the importance of checking for any potential errors or pitfalls that may arise during the simplification process.

The Given Expression

The given expression is:

$$\frac{x+7}{7x+35} \cdot \frac{x^2-3x-40}{x-8}$$

Our goal is to simplify this expression by finding an equivalent expression that is easier to work with.

Step 1: Factor the Numerators

To simplify the expression, we need to factor the numerators of both fractions. Let's start with the first numerator, $x+7$. We can factor this expression as:

$$x+7 = (x+7)$$

The second numerator, $x^2-3x-40$, can be factored as:

$$x^2-3x-40 = (x-8)(x+5)$$

Step 2: Factor the Denominators

Next, we need to factor the denominators of both fractions. The first denominator, $7x+35$, can be factored as:

$$7x+35 = 7(x+5)$$

The second denominator, $x-8$, is already factored.

Step 3: Cancel Out Common Factors

Now that we have factored the numerators and denominators, we can cancel out any common factors. In this case, we can cancel out the $(x+5)$ factor in the numerator and denominator of the first fraction, as well as the $(x-8)$ factor in the numerator and denominator of the second fraction.

After canceling out the common factors, the expression becomes:

$$\frac{1}{7} \cdot \frac{x+5}{x-8}$$

Step 4: Simplify the Expression

Finally, we can simplify the expression by multiplying the two fractions together. This gives us:

$$\frac{x+5}{7(x-8)}$$

Conclusion

In this article, we have simplified the given algebraic expression by factoring and canceling out common factors. We have also discussed the importance of checking for any potential errors or pitfalls that may arise during the simplification process.

The Correct Answer

Based on our simplification, the correct answer is:

A. $\frac{x+5}{7(x-8)}$

Discussion

The given expression can be simplified by factoring and canceling out common factors. This process involves breaking down the expression into smaller parts, factoring the numerators and denominators, and then canceling out any common factors. By following these steps, we can simplify the expression and arrive at the correct answer.

Tips and Tricks

When simplifying algebraic expressions, it's essential to check for any potential errors or pitfalls that may arise during the simplification process. Some common pitfalls include:

• Not factoring the numerators and denominators correctly
• Not canceling out common factors correctly
• Not checking for any potential errors or pitfalls

By being aware of these potential pitfalls, we can avoid making mistakes and arrive at the correct answer.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications. For example, in physics, algebraic expressions are used to describe the motion of objects. By simplifying these expressions, physicists can better understand the behavior of objects and make more accurate predictions.

In engineering, algebraic expressions are used to design and optimize systems. By simplifying these expressions, engineers can create more efficient and effective systems.

Conclusion

Introduction

In our previous article, we explored how to simplify algebraic expressions by factoring and canceling out common factors. In this article, we will answer some of the most frequently asked questions about simplifying algebraic expressions.

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to factor the numerators and denominators. This involves breaking down the expression into smaller parts and identifying any common factors.

Q: How do I factor the numerators and denominators?

A: To factor the numerators and denominators, you need to identify any common factors and break them down into smaller parts. For example, if you have the expression $\frac{x+7}{7x+35}$, you can factor the numerator as $(x+7)$ and the denominator as $7(x+5)$.

Q: What is the difference between factoring and canceling out common factors?

A: Factoring involves breaking down an expression into smaller parts, while canceling out common factors involves eliminating any common factors between the numerator and denominator.

Q: How do I know when to cancel out common factors?

A: You should cancel out common factors when you have identified any common factors between the numerator and denominator. This will help simplify the expression and make it easier to work with.

Q: What are some common pitfalls to avoid when simplifying algebraic expressions?

A: Some common pitfalls to avoid when simplifying algebraic expressions include:

• Not factoring the numerators and denominators correctly
• Not canceling out common factors correctly
• Not checking for any potential errors or pitfalls

Q: How do I check for potential errors or pitfalls?

A: To check for potential errors or pitfalls, you should:

• Double-check your factoring and canceling out common factors
• Verify that you have not missed any common factors
• Check your work for any potential errors or mistakes

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has numerous real-world applications, including:

• Physics: Algebraic expressions are used to describe the motion of objects. By simplifying these expressions, physicists can better understand the behavior of objects and make more accurate predictions.
• Engineering: Algebraic expressions are used to design and optimize systems. By simplifying these expressions, engineers can create more efficient and effective systems.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by:

• Working through example problems
• Practicing with different types of expressions
• Using online resources or worksheets to help you practice

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill to master in mathematics. By following the steps outlined in this article and practicing regularly, you can become proficient in simplifying algebraic expressions and apply this skill to real-world problems.

Additional Resources

For more information on simplifying algebraic expressions, you can check out the following resources:

• Khan Academy: Algebraic Expressions
• Mathway: Simplifying Algebraic Expressions
• Wolfram Alpha: Algebraic Expressions

Final Tips

Remember to:

• Double-check your work
• Verify that you have not missed any common factors
• Practice regularly to become proficient in simplifying algebraic expressions

By following these tips and practicing regularly, you can become a master of simplifying algebraic expressions and apply this skill to real-world problems.