Express $9.2 - 3 \sqrt{-8}$ As A Complex Number In The Form Of $a + Bi$. What Is The Imaginary Term?A. \$-i \sqrt{2}$[/tex\] B. $6i$ C. $-6i$ D. $-6i \sqrt{2}$

Feb 28, 2025 by ADMIN 176 views

$**Solving Complex Numbers: Expressing $9.2 - 3 \sqrt{-8}$ as a Complex Number**$

Introduction

In mathematics, complex numbers are a fundamental concept that extends the real number system to include numbers with both real and imaginary parts. The imaginary unit, denoted by $i$ , is defined as the square root of $-1$ . In this article, we will explore how to express the given expression $9.2 - 3 \sqrt{-8}$ as a complex number in the form of $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

Understanding Complex Numbers

A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit. The real part of the complex number is $a$ , and the imaginary part is $b$ . Complex numbers can be added, subtracted, multiplied, and divided just like real numbers.

Simplifying the Expression

To simplify the expression $9.2 - 3 \sqrt{-8}$, we need to start by simplifying the square root term. We can rewrite $\sqrt{-8}$ as $\sqrt{-1} \cdot \sqrt{8}$. Since $\sqrt{-1} = i$, we can rewrite the expression as $9.2 - 3i\sqrt{8}$.

Simplifying the Square Root Term

Now, let's simplify the square root term $\sqrt{8}$. We can rewrite $\sqrt{8}$ as $\sqrt{4 \cdot 2}$. Using the property of square roots, we can rewrite this as $\sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$.

Substituting the Simplified Square Root Term

Now, let's substitute the simplified square root term back into the expression. We have $9.2 - 3i\sqrt{8} = 9.2 - 3i(2\sqrt{2})$.

Simplifying the Expression Further

Now, let's simplify the expression further. We can rewrite $9.2 - 3i(2\sqrt{2})$ as $9.2 - 6i\sqrt{2}$.

Expressing the Complex Number in the Form $a + bi$

Now, let's express the complex number in the form $a + bi$. We have $a = 9.2$ and $b = -6\sqrt{2}$.

Conclusion

In this article, we have explored how to express the given expression $9.2 - 3 \sqrt{-8}$ as a complex number in the form of $a + bi$. We simplified the square root term and substituted it back into the expression. Finally, we expressed the complex number in the form $a + bi$. The imaginary term of the complex number is $-6i\sqrt{2}$.

Answer

The imaginary term of the complex number is $-6i\sqrt{2}$.

Discussion

This problem requires a good understanding of complex numbers and their properties. The student needs to be able to simplify the square root term and substitute it back into the expression. The student also needs to be able to express the complex number in the form $a + bi$. This problem is a good exercise for students who are learning about complex numbers.

References

[1] "Complex Numbers" by Math Open Reference. Retrieved from https://www.mathopenref.com/complexnumbers.html
[2] "Complex Numbers" by Khan Academy. Retrieved from https://www.khanacademy.org/math/linear-algebra/complex-numbers

Keywords

Complex numbers
Imaginary unit
Square root of -1
Simplifying complex expressions
Expressing complex numbers in the form $a + bi$
Frequently Asked Questions (FAQs) about Complex Numbers ===========================================================

Q: What is a complex number?

A: A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

Q: What is the imaginary unit?

A: The imaginary unit, denoted by $i$ , is defined as the square root of $-1$ . It is used to extend the real number system to include numbers with both real and imaginary parts.

Q: How do I simplify a complex expression?

A: To simplify a complex expression, you need to start by simplifying the square root term. You can rewrite $\sqrt{-8}$ as $\sqrt{-1} \cdot \sqrt{8}$. Since $\sqrt{-1} = i$, you can rewrite the expression as $9.2 - 3i\sqrt{8}$.

Q: How do I simplify the square root term?

A: To simplify the square root term, you can rewrite $\sqrt{8}$ as $\sqrt{4 \cdot 2}$. Using the property of square roots, you can rewrite this as $\sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}$.

Q: How do I express a complex number in the form $a + bi$?

A: To express a complex number in the form $a + bi$, you need to identify the real part $a$ and the imaginary part $b$ . In the expression $9.2 - 3i\sqrt{8}$, the real part is $9.2$ and the imaginary part is $-6\sqrt{2}$.

Q: What is the imaginary term of the complex number?

A: The imaginary term of the complex number is $-6i\sqrt{2}$.

Q: How do I add, subtract, multiply, and divide complex numbers?

A: Complex numbers can be added, subtracted, multiplied, and divided just like real numbers. However, when multiplying and dividing complex numbers, you need to use the distributive property and the fact that $i^2 = -1$.

Q: What are some common mistakes to avoid when working with complex numbers?

A: Some common mistakes to avoid when working with complex numbers include:

Forgetting to simplify the square root term
Not using the distributive property when multiplying and dividing complex numbers
Not using the fact that $i^2 = -1$ when multiplying and dividing complex numbers
Not expressing complex numbers in the form $a + bi$

Q: What are some real-world applications of complex numbers?

A: Complex numbers have many real-world applications, including:

Electrical engineering: Complex numbers are used to represent AC circuits and analyze their behavior.
Signal processing: Complex numbers are used to represent signals and analyze their frequency content.
Control systems: Complex numbers are used to analyze and design control systems.
Quantum mechanics: Complex numbers are used to represent wave functions and analyze the behavior of particles.

Q: What are some resources for learning more about complex numbers?

A: Some resources for learning more about complex numbers include:

Online tutorials and videos
Textbooks and reference books
Online forums and communities
Online courses and certification programs

Conclusion

In this article, we have answered some frequently asked questions about complex numbers. We have covered topics such as simplifying complex expressions, expressing complex numbers in the form $a + bi$, and adding, subtracting, multiplying, and dividing complex numbers. We have also discussed some common mistakes to avoid and some real-world applications of complex numbers. Finally, we have provided some resources for learning more about complex numbers.