Simplify \[$(8+3i)-(6-2i)\$\].A. \[$14+i\$\] B. \[$2+i\$\] C. \[$14+5i\$\] D. \[$2+5i\$\]
Introduction
Complex numbers are an essential part of mathematics, and they have numerous applications in various fields, including engineering, physics, and computer science. In this article, we will focus on simplifying complex numbers, which is a crucial operation in mathematics. We will use the given expression {(8+3i)-(6-2i)$}$ as an example to demonstrate the steps involved in simplifying complex numbers.
What are Complex Numbers?
Complex numbers are numbers 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$}$. The real part of a complex number is {a$}$, and the imaginary part is {bi$}$.
Simplifying Complex Numbers
To simplify a complex number, we need to combine the real and imaginary parts. Let's consider the given expression {(8+3i)-(6-2i)$}$. To simplify this expression, we need to follow the order of operations (PEMDAS):
- Parentheses: Evaluate the expressions inside the parentheses.
- Exponents: Evaluate any exponents (none in this case).
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Step 1: Evaluate the Expressions Inside the Parentheses
The given expression is {(8+3i)-(6-2i)$}$. To evaluate the expressions inside the parentheses, we need to follow the order of operations:
{(8+3i)-(6-2i)$}$ = {(8+3i)-6+2i$}$
Step 2: Combine the Real and Imaginary Parts
Now, let's combine the real and imaginary parts:
{(8+3i)-6+2i$}$ = ${8-6+3i+2i\$}
Step 3: Simplify the Real and Imaginary Parts
Now, let's simplify the real and imaginary parts:
${8-6+3i+2i\$} = ${2+5i\$}
Conclusion
In this article, we have demonstrated the steps involved in simplifying complex numbers. We used the given expression {(8+3i)-(6-2i)$}$ as an example to show how to combine the real and imaginary parts. The simplified expression is ${2+5i\$}. This is the correct answer among the given options.
Answer
The correct answer is:
- D. ${2+5i\$}
Why is this the Correct Answer?
This is the correct answer because we have followed the order of operations and combined the real and imaginary parts correctly. The expression {(8+3i)-(6-2i)$}$ simplifies to ${2+5i\$}, which is the correct answer.
What are the Other Options?
The other options are:
- A. ${14+i\$}
- B. ${2+i\$}
- C. ${14+5i\$}
These options are incorrect because they do not follow the order of operations and do not combine the real and imaginary parts correctly.
Conclusion
Introduction
In our previous article, we demonstrated the steps involved in simplifying complex numbers using the given expression {(8+3i)-(6-2i)$}$ as an example. In this article, we will provide a Q&A guide to help you understand the concept of simplifying complex numbers better.
Q: What are Complex Numbers?
A: Complex numbers are numbers 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$}$.
Q: What is the Imaginary Unit?
A: The imaginary unit, denoted by {i$}$, is a mathematical concept that satisfies the equation {i^2=-1$}$. It is used to extend the real number system to the complex number system.
Q: How Do I Simplify Complex Numbers?
A: To simplify a complex number, you need to follow the order of operations (PEMDAS):
- Parentheses: Evaluate the expressions inside the parentheses.
- Exponents: Evaluate any exponents (none in this case).
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: What is the Order of Operations?
A: The order of operations is a set of rules that tells you which operations to perform first when you have multiple operations in an expression. The order of operations is:
- Parentheses: Evaluate the expressions inside the parentheses.
- Exponents: Evaluate any exponents (none in this case).
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: How Do I Combine the Real and Imaginary Parts?
A: To combine the real and imaginary parts, you need to add or subtract the real parts and add or subtract the imaginary parts separately.
Q: What is the Difference Between Real and Imaginary Parts?
A: The real part of a complex number is the part that is not multiplied by the imaginary unit {i$}$. The imaginary part of a complex number is the part that is multiplied by the imaginary unit {i$}$.
Q: How Do I Simplify the Real and Imaginary Parts?
A: To simplify the real and imaginary parts, you need to combine the real parts and combine the imaginary parts separately.
Q: What is the Final Answer?
A: The final answer is ${2+5i\$}, which is the correct answer among the given options.
Conclusion
In conclusion, simplifying complex numbers is an essential operation in mathematics. We have provided a Q&A guide to help you understand the concept of simplifying complex numbers better. By following the order of operations and combining the real and imaginary parts correctly, you can simplify complex numbers and arrive at the correct answer.
Frequently Asked Questions
- Q: What is the difference between real and imaginary parts? A: The real part of a complex number is the part that is not multiplied by the imaginary unit {i$}$. The imaginary part of a complex number is the part that is multiplied by the imaginary unit {i$}$.
- Q: How do I simplify complex numbers?
A: To simplify a complex number, you need to follow the order of operations (PEMDAS):
- Parentheses: Evaluate the expressions inside the parentheses.
- Exponents: Evaluate any exponents (none in this case).
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
- Q: What is the final answer? A: The final answer is ${2+5i\$}, which is the correct answer among the given options.
Common Mistakes
- Mistake 1: Not following the order of operations correctly.
- Mistake 2: Not combining the real and imaginary parts correctly.
- Mistake 3: Not simplifying the real and imaginary parts correctly.
Conclusion
In conclusion, simplifying complex numbers is an essential operation in mathematics. By following the order of operations and combining the real and imaginary parts correctly, you can simplify complex numbers and arrive at the correct answer.