Simplify: 1/(x - 5) + (-1)/(x + 5) A) 10/(x² - 25) B) -10/(x² - 25) C) 10/(x² + 25) D) -10/(x² + 25)
Introduction
Complex fractions can be a daunting task for many students and mathematicians alike. However, with the right approach and techniques, simplifying complex fractions can be a breeze. In this article, we will explore the process of simplifying complex fractions, using the example of 1/(x - 5) + (-1)/(x + 5) as our case study.
Understanding Complex Fractions
A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In the given example, we have two fractions: 1/(x - 5) and (-1)/(x + 5). To simplify this expression, we need to find a common denominator and combine the fractions.
Finding a Common Denominator
The first step in simplifying complex fractions is to find a common denominator. In this case, the denominators are (x - 5) and (x + 5). To find a common denominator, we need to multiply the two denominators together: (x - 5)(x + 5).
Multiplying the Denominators
When we multiply the denominators, we get:
(x - 5)(x + 5) = x² - 25
This is the common denominator that we will use to simplify the complex fraction.
Rewriting the Complex Fraction
Now that we have found the common denominator, we can rewrite the complex fraction as:
1/(x - 5) + (-1)/(x + 5) = (1(x + 5) + (-1)(x - 5)) / (x² - 25)
Simplifying the Numerator
The next step is to simplify the numerator. We can do this by combining like terms:
(1(x + 5) + (-1)(x - 5)) = x + 5 - x + 5 = 10
So, the numerator simplifies to 10.
Simplifying the Complex Fraction
Now that we have simplified the numerator, we can rewrite the complex fraction as:
10 / (x² - 25)
Conclusion
In this article, we have explored the process of simplifying complex fractions using the example of 1/(x - 5) + (-1)/(x + 5). We found a common denominator, multiplied the denominators together, and simplified the numerator. The final simplified expression is 10 / (x² - 25).
Answer
The correct answer is:
A) 10/(x² - 25)
Discussion
This problem is a great example of how to simplify complex fractions. By following the steps outlined in this article, you can simplify even the most complex fractions. Remember to always find a common denominator, multiply the denominators together, and simplify the numerator.
Common Mistakes
When simplifying complex fractions, it's easy to make mistakes. Here are a few common mistakes to watch out for:
- Not finding a common denominator
- Not multiplying the denominators together
- Not simplifying the numerator
Tips and Tricks
Here are a few tips and tricks to help you simplify complex fractions:
- Always start by finding a common denominator
- Multiply the denominators to get the common denominator
- Simplify the numerator by combining like terms
- Check your work by plugging in values for x
Real-World Applications
Simplifying complex fractions has many real-world applications. For example, in physics, complex fractions are used to describe the motion of objects. In engineering, complex fractions are used to design and analyze systems. In finance, complex fractions are used to calculate interest rates and investment returns.
Conclusion
Introduction
In our previous article, we explored the process of simplifying complex fractions using the example of 1/(x - 5) + (-1)/(x + 5). In this article, we will answer some of the most frequently asked questions about simplifying complex fractions.
Q: What is a complex fraction?
A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.
Q: Why do I need to find a common denominator?
A: Finding a common denominator is essential when simplifying complex fractions. It allows you to combine the fractions and simplify the expression.
Q: How do I find a common denominator?
A: To find a common denominator, you need to multiply the two denominators together. In the case of 1/(x - 5) + (-1)/(x + 5), the common denominator is (x - 5)(x + 5) = x² - 25.
Q: What if I have a fraction with a variable in the denominator?
A: If you have a fraction with a variable in the denominator, you can use the same process as before. However, you may need to use algebraic techniques to simplify the expression.
Q: Can I simplify complex fractions with negative exponents?
A: Yes, you can simplify complex fractions with negative exponents. To do this, you need to use the rule that a⁻¹ = 1/a.
Q: How do I simplify complex fractions with multiple fractions in the numerator?
A: To simplify complex fractions with multiple fractions in the numerator, you need to combine the fractions and simplify the expression. You can do this by finding a common denominator and combining the fractions.
Q: What if I have a complex fraction with a fraction in the denominator?
A: If you have a complex fraction with a fraction in the denominator, you can use the same process as before. However, you may need to use algebraic techniques to simplify the expression.
Q: Can I simplify complex fractions with radicals in the denominator?
A: Yes, you can simplify complex fractions with radicals in the denominator. To do this, you need to use the rule that √a = a^(1/2).
Q: How do I check my work when simplifying complex fractions?
A: To check your work, you can plug in values for the variable and simplify the expression. You can also use algebraic techniques to verify your answer.
Q: What are some common mistakes to watch out for when simplifying complex fractions?
A: Some common mistakes to watch out for when simplifying complex fractions include:
- Not finding a common denominator
- Not multiplying the denominators together
- Not simplifying the numerator
- Not checking your work
Q: How can I practice simplifying complex fractions?
A: You can practice simplifying complex fractions by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.
Conclusion
In conclusion, simplifying complex fractions is a crucial skill that has many real-world applications. By following the steps outlined in this article and practicing regularly, you can become proficient in simplifying complex fractions. Remember to always find a common denominator, multiply the denominators together, and simplify the numerator. With practice and patience, you will become a master of simplifying complex fractions.