Simplify The Expression Below As Much As Possible. { (7-i)+(5+3i)-(-3+2i)$}$A. ${ 15-4i\$} B. 15 C. 9 D. ${ 9-4i\$}

Introduction

Complex expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will focus on simplifying the given expression: {(7-i)+(5+3i)-(-3+2i)$}$. We will break down the steps involved in simplifying complex expressions and provide a clear explanation of each step.

Understanding Complex Numbers

Before we dive into simplifying the expression, let's briefly review 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, which satisfies the equation: ${i^2 = -1\$}$.

Step 1: Distribute the Negative Sign

The first step in simplifying the expression is to distribute the negative sign to the terms inside the parentheses: ${-(-3+2i) = 3-2i\$}$.

(7-i) + (5+3i) - (-3+2i)
= (7-i) + (5+3i) + (3-2i)

Step 2: Combine Like Terms

Now, let's combine the like terms: ${7+5+3 = 15}$ and ${-i+3i-2i = 0i\$}$.

= 15 + 0i

Step 3: Simplify the Expression

The expression is now simplified to: ${15\$}$. However, we need to consider the options provided and determine if the expression can be simplified further.

Analyzing the Options

Let's analyze the options provided:

• A. ${15-4i\$}
• B. 15
• C. 9
• D. ${9-4i\$}

Based on our simplification, option B is the correct answer.

Conclusion

Simplifying complex expressions requires a step-by-step approach. By distributing the negative sign, combining like terms, and analyzing the options, we can simplify the expression: {(7-i)+(5+3i)-(-3+2i)$}$ to ${15\$}$. This article has provided a clear explanation of each step involved in simplifying complex expressions, making it easier for students and professionals to understand and apply this concept.

Additional Tips and Resources

• To simplify complex expressions, it's essential to follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
• When combining like terms, make sure to add or subtract the coefficients of the terms with the same variable.
• For more practice and examples, check out online resources such as Khan Academy, Mathway, or Wolfram Alpha.

Final Answer

Introduction

In our previous article, we explored the concept of simplifying complex expressions and provided a step-by-step guide on how to simplify the expression: {(7-i)+(5+3i)-(-3+2i)$}$. In this article, we will address some of the most frequently asked questions related to simplifying complex expressions.

Q&A

Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be expressed without any imaginary unit, whereas an imaginary number is a number that can be expressed with the imaginary unit, ${i\$. For example, \[3}$ is a real number, while ${3i}$ is an imaginary number.

Q: How do I simplify a complex expression with multiple terms?

A: To simplify a complex expression with multiple terms, follow these steps:

1. Distribute the negative sign to the terms inside the parentheses.
2. Combine like terms by adding or subtracting the coefficients of the terms with the same variable.
3. Simplify the expression by combining the real and imaginary parts.

Q: What is the order of operations for simplifying complex expressions?

A: The order of operations for simplifying complex expressions is:

1. Parentheses: Evaluate the expressions inside the parentheses first.
2. Exponents: Evaluate any exponents next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I handle complex expressions with fractions?

A: To handle complex expressions with fractions, follow these steps:

1. Multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary unit.
2. Simplify the expression by combining the real and imaginary parts.
3. Reduce the fraction to its simplest form.

Q: What are some common mistakes to avoid when simplifying complex expressions?

A: Some common mistakes to avoid when simplifying complex expressions include:

• Forgetting to distribute the negative sign to the terms inside the parentheses.
• Failing to combine like terms.
• Not simplifying the expression by combining the real and imaginary parts.
• Not following the order of operations.

Q: How can I practice simplifying complex expressions?

A: To practice simplifying complex expressions, try the following:

• Start with simple expressions and gradually move on to more complex ones.
• Use online resources such as Khan Academy, Mathway, or Wolfram Alpha to practice and get instant feedback.
• Work with a partner or join a study group to practice and learn from others.
• Take online quizzes or tests to assess your knowledge and identify areas for improvement.

Conclusion

Simplifying complex expressions is a crucial skill for students and professionals alike. By following the steps outlined in this article and avoiding common mistakes, you can simplify complex expressions with confidence. Remember to practice regularly and seek help when needed to become proficient in simplifying complex expressions.

Additional Resources

• Khan Academy: Complex Numbers
• Mathway: Complex Numbers
• Wolfram Alpha: Complex Numbers
• MIT OpenCourseWare: Complex Analysis

Final Tips

• Simplifying complex expressions requires patience and practice.
• Don't be afraid to ask for help when needed.
• Use online resources to practice and get instant feedback.
• Join a study group or work with a partner to learn from others.

Final Answer

The final answer is: There is no final answer, as this article is a Q&A guide.