Divide $\frac{5i}{2+5i}$.Write Your Answer As A Complex Number In Standard Form.

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Introduction

When it comes to dividing complex fractions, it can be a daunting task, especially for those who are new to the concept. However, with a clear understanding of the steps involved and a bit of practice, you'll be able to simplify complex fractions like $\frac{5i}{2+5i}$ in no time. In this article, we'll take a closer look at how to divide complex fractions and provide a step-by-step guide on how to simplify the given expression.

Understanding Complex Fractions

Before we dive into the steps involved in dividing complex fractions, let's first understand what complex fractions are. A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In other words, it's a fraction that has a fraction as one of its components. Complex fractions can be written in the form $\frac{a}{b}$, where $a$ and $b$ are both fractions.

The Steps Involved in Dividing Complex Fractions

When it comes to dividing complex fractions, there are several steps involved. Here are the steps you need to follow:

1. Multiply the numerator and denominator by the conjugate of the denominator: The conjugate of a complex number is obtained by changing the sign of the imaginary part. For example, the conjugate of $2+5i$ is $2-5i$. Multiplying the numerator and denominator by the conjugate of the denominator helps to eliminate the imaginary part from the denominator.
2. Simplify the expression: Once you've multiplied the numerator and denominator by the conjugate of the denominator, simplify the expression by combining like terms.
3. Write the answer in standard form: The final step is to write the answer in standard form, which is the form $a+bi$, where $a$ and $b$ are real numbers.

Dividing $\frac{5i}{2+5i}$: A Step-by-Step Guide

Now that we've covered the steps involved in dividing complex fractions, let's apply these steps to the given expression $\frac{5i}{2+5i}$.

Step 1: Multiply the numerator and denominator by the conjugate of the denominator

To eliminate the imaginary part from the denominator, we need to multiply the numerator and denominator by the conjugate of $2+5i$, which is $2-5i$. This gives us:

$$\frac{5i}{2+5i} \cdot \frac{2-5i}{2-5i}$$

Step 2: Simplify the expression

Now that we've multiplied the numerator and denominator by the conjugate of the denominator, let's simplify the expression:

$$\frac{5i(2-5i)}{(2+5i)(2-5i)}$$

Expanding the numerator and denominator, we get:

$$\frac{10i-25i^2}{4-25i^2}$$

Since $i^2 = -1$, we can substitute this value into the expression:

$$\frac{10i-25(-1)}{4-25(-1)}$$

Simplifying further, we get:

$$\frac{10i+25}{4+25}$$

Step 3: Write the answer in standard form

The final step is to write the answer in standard form, which is the form $a+bi$, where $a$ and $b$ are real numbers. In this case, the answer is:

$$\frac{10i+25}{29}$$

However, we can simplify this expression further by separating the real and imaginary parts:

$$\frac{25}{29} + \frac{10i}{29}$$

Conclusion

Dividing complex fractions can be a challenging task, but with a clear understanding of the steps involved and a bit of practice, you'll be able to simplify complex fractions like $\frac{5i}{2+5i}$ in no time. By following the steps outlined in this article, you'll be able to divide complex fractions and write the answer in standard form.

Frequently Asked Questions

• What is a complex fraction? A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.
• How do I divide complex fractions? To divide complex fractions, multiply the numerator and denominator by the conjugate of the denominator, simplify the expression, and write the answer in standard form.
• What is the conjugate of a complex number? The conjugate of a complex number is obtained by changing the sign of the imaginary part.

Additional Resources

• Complex Fractions: A Step-by-Step Guide This article provides a comprehensive guide to complex fractions, including how to divide them and write the answer in standard form.
• Dividing Complex Fractions: A Video Tutorial This video tutorial provides a step-by-step guide to dividing complex fractions and writing the answer in standard form.
• Complex Fractions: Practice Problems This article provides practice problems to help you master the concept of complex fractions and dividing them.
  

Introduction

Complex fractions can be a challenging topic to understand, especially when it comes to dividing them. However, with a clear understanding of the concepts and a bit of practice, you'll be able to simplify complex fractions like a pro. In this article, we'll answer some of the most frequently asked questions about complex fractions and provide a comprehensive guide to help you master this topic.

Q&A: 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: How do I divide complex fractions?

A: To divide complex fractions, multiply the numerator and denominator by the conjugate of the denominator, simplify the expression, and write the answer in standard form.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number is obtained by changing the sign of the imaginary part.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, multiply the numerator and denominator by the conjugate of the denominator, simplify the expression, and write the answer in standard form.

Q: What is the standard form of a complex fraction?

A: The standard form of a complex fraction is the form $a+bi$, where $a$ and $b$ are real numbers.

Q: How do I write a complex fraction in standard form?

A: To write a complex fraction in standard form, simplify the expression by combining like terms and separating the real and imaginary parts.

Q: What are some common mistakes to avoid when dividing complex fractions?

A: Some common mistakes to avoid when dividing complex fractions include:

• Not multiplying the numerator and denominator by the conjugate of the denominator
• Not simplifying the expression
• Not writing the answer in standard form

Q: How do I practice dividing complex fractions?

A: To practice dividing complex fractions, try the following:

• Start with simple complex fractions and gradually move on to more complex ones
• Use online resources or practice problems to help you master the concept
• Practice dividing complex fractions regularly to build your confidence and skills

Additional Resources

• Complex Fractions: A Step-by-Step Guide This article provides a comprehensive guide to complex fractions, including how to divide them and write the answer in standard form.
• Dividing Complex Fractions: A Video Tutorial This video tutorial provides a step-by-step guide to dividing complex fractions and writing the answer in standard form.
• Complex Fractions: Practice Problems This article provides practice problems to help you master the concept of complex fractions and dividing them.

Conclusion

Dividing complex fractions can be a challenging task, but with a clear understanding of the concepts and a bit of practice, you'll be able to simplify complex fractions like a pro. By following the steps outlined in this article and practicing regularly, you'll be able to master the concept of complex fractions and divide them with confidence.

Frequently Asked Questions

• What is a complex fraction? A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.
• How do I divide complex fractions? To divide complex fractions, multiply the numerator and denominator by the conjugate of the denominator, simplify the expression, and write the answer in standard form.
• What is the conjugate of a complex number? The conjugate of a complex number is obtained by changing the sign of the imaginary part.

Related Articles

• Complex Fractions: A Step-by-Step Guide This article provides a comprehensive guide to complex fractions, including how to divide them and write the answer in standard form.
• Dividing Complex Fractions: A Video Tutorial This video tutorial provides a step-by-step guide to dividing complex fractions and writing the answer in standard form.
• Complex Fractions: Practice Problems This article provides practice problems to help you master the concept of complex fractions and dividing them.