Divide And Express The Result In Standard Form.${ \frac{6}{7-i} }$ { \frac{6}{7-i} = \square \} (Simplify Your Answer. Type Your Answer In The Form { A + Bi$}$.)

Mar 10, 2025 by ADMIN 164 views

Rationalizing the Denominator: A Step-by-Step Guide to Simplifying Complex Fractions

In mathematics, rationalizing the denominator is a crucial technique used to simplify complex fractions. This process involves eliminating any radicals or imaginary numbers from the denominator, making it easier to work with and understand the expression. In this article, we will explore the concept of rationalizing the denominator and provide a step-by-step guide on how to simplify complex fractions.

What is Rationalizing the Denominator?

Rationalizing the denominator is a mathematical process that involves multiplying both the numerator and the denominator of a fraction by a specific value, usually a conjugate, to eliminate any radicals or imaginary numbers from the denominator. This process is essential in simplifying complex fractions and making them easier to work with.

Why is Rationalizing the Denominator Important?

Rationalizing the denominator is an important concept in mathematics because it allows us to simplify complex fractions and make them easier to work with. By eliminating any radicals or imaginary numbers from the denominator, we can perform calculations and operations more easily, making it a crucial technique in various mathematical applications.

Step-by-Step Guide to Rationalizing the Denominator

To rationalize the denominator, follow these steps:

Identify the Conjugate: The first step in rationalizing the denominator is to identify the conjugate of the denominator. The conjugate of a binomial expression is obtained by changing the sign of the second term. For example, the conjugate of $7-i$ is $7+i$ .
Multiply the Numerator and Denominator: Multiply both the numerator and the denominator by the conjugate of the denominator. In this case, we multiply both the numerator and the denominator by $7+i$ .
Simplify the Expression: Simplify the expression by multiplying the numerator and the denominator. This will eliminate any radicals or imaginary numbers from the denominator.
Write the Final Answer: Write the final answer in the form $a+bi$ , where $a$ and $b$ are real numbers.

Example: Rationalizing the Denominator

Let's consider the complex fraction $\frac{6}{7-i}$ . To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator, which is $7+i$ .

$\frac{6}{7-i} = \frac{6(7+i)}{(7-i)(7+i)}$

Expanding the denominator, we get:

$\frac{6(7+i)}{49-i^2}$

Since $i^2 = -1$ , we can simplify the denominator:

$\frac{6(7+i)}{49+1}$

Combining the terms in the denominator, we get:

$\frac{6(7+i)}{50}$

Simplifying the expression, we get:

$\frac{42+6i}{50}$

Writing the final answer in the form $a+bi$ , we get:

$\frac{6}{5} + \frac{3}{25}i$

Conclusion

Rationalizing the denominator is a crucial technique used to simplify complex fractions. By following the step-by-step guide outlined in this article, you can easily rationalize the denominator and simplify complex fractions. Remember to identify the conjugate, multiply the numerator and denominator, simplify the expression, and write the final answer in the form $a+bi$ . With practice, you will become proficient in rationalizing the denominator and simplifying complex fractions.

Common Mistakes to Avoid

When rationalizing the denominator, there are several common mistakes to avoid:

Not identifying the conjugate: Make sure to identify the conjugate of the denominator before multiplying the numerator and denominator.
Not multiplying the numerator and denominator: Remember to multiply both the numerator and the denominator by the conjugate of the denominator.
Not simplifying the expression: Simplify the expression by combining like terms and eliminating any radicals or imaginary numbers from the denominator.
Not writing the final answer in the correct form: Write the final answer in the form $a+bi$ , where $a$ and $b$ are real numbers.

Real-World Applications

Rationalizing the denominator has several real-world applications in various fields, including:

Engineering: Rationalizing the denominator is used to simplify complex expressions in electrical engineering, mechanical engineering, and other fields.
Physics: Rationalizing the denominator is used to simplify complex expressions in physics, particularly in the study of waves and vibrations.
Computer Science: Rationalizing the denominator is used to simplify complex expressions in computer science, particularly in the study of algorithms and data structures.

Conclusion

In conclusion, rationalizing the denominator is a crucial technique used to simplify complex fractions. By following the step-by-step guide outlined in this article, you can easily rationalize the denominator and simplify complex fractions. Remember to identify the conjugate, multiply the numerator and denominator, simplify the expression, and write the final answer in the form $a+bi$ . With practice, you will become proficient in rationalizing the denominator and simplifying complex fractions.
Rationalizing the Denominator: A Q&A Guide

In our previous article, we explored the concept of rationalizing the denominator and provided a step-by-step guide on how to simplify complex fractions. However, we understand that there may be some questions and concerns that you may have. In this article, we will address some of the most frequently asked questions about rationalizing the denominator.

Q: What is the purpose of rationalizing the denominator?

A: The purpose of rationalizing the denominator is to eliminate any radicals or imaginary numbers from the denominator, making it easier to work with and understand the expression.

Q: How do I identify the conjugate of a binomial expression?

A: To identify the conjugate of a binomial expression, simply change the sign of the second term. For example, the conjugate of $7-i$ is $7+i$ .

Q: Why do I need to multiply the numerator and denominator by the conjugate?

A: Multiplying the numerator and denominator by the conjugate eliminates any radicals or imaginary numbers from the denominator, making it easier to simplify the expression.

Q: Can I rationalize the denominator of a fraction with a rational denominator?

A: No, you cannot rationalize the denominator of a fraction with a rational denominator. Rationalizing the denominator is only necessary when the denominator contains radicals or imaginary numbers.

Q: How do I simplify the expression after rationalizing the denominator?

A: To simplify the expression, combine like terms and eliminate any radicals or imaginary numbers from the denominator.

Q: What is the final answer format for rationalizing the denominator?

A: The final answer format for rationalizing the denominator is $a+bi$ , where $a$ and $b$ are real numbers.

Q: Can I use a calculator to rationalize the denominator?

A: Yes, you can use a calculator to rationalize the denominator. However, it's essential to understand the concept and process of rationalizing the denominator to ensure accuracy and precision.

Q: How do I apply rationalizing the denominator in real-world applications?

A: Rationalizing the denominator is used in various real-world applications, including engineering, physics, and computer science. It's essential to understand the concept and process of rationalizing the denominator to apply it in these fields.

Q: What are some common mistakes to avoid when rationalizing the denominator?

A: Some common mistakes to avoid when rationalizing the denominator include:

Not identifying the conjugate
Not multiplying the numerator and denominator
Not simplifying the expression
Not writing the final answer in the correct format

Q: Can I rationalize the denominator of a fraction with a complex numerator?

A: Yes, you can rationalize the denominator of a fraction with a complex numerator. However, it's essential to follow the same process and steps as rationalizing the denominator of a fraction with a rational numerator.

Q: How do I check my work when rationalizing the denominator?

A: To check your work, simplify the expression and ensure that the denominator is rational. You can also use a calculator to verify the result.

Conclusion

In conclusion, rationalizing the denominator is a crucial technique used to simplify complex fractions. By understanding the concept and process of rationalizing the denominator, you can apply it in various real-world applications and simplify complex expressions with ease. Remember to identify the conjugate, multiply the numerator and denominator, simplify the expression, and write the final answer in the correct format. With practice, you will become proficient in rationalizing the denominator and simplifying complex fractions.

Additional Resources

For further learning and practice, we recommend the following resources:

Khan Academy: Rationalizing the Denominator
Mathway: Rationalizing the Denominator
Wolfram Alpha: Rationalizing the Denominator

Practice Problems

To practice rationalizing the denominator, try the following problems:

Rationalize the denominator of $\frac{3}{2-i}$ .
Rationalize the denominator of $\frac{4}{3+2i}$ .
Rationalize the denominator of $\frac{5}{1-3i}$ .

Answer Key

$\frac{3(2+i)}{5}$
$\frac{4(3-2i)}{13}$
$\frac{5(1+3i)}{10}$