Simplify The Expression { (8-5i) - (-2+3i)$}$.A. ${ 10-8i\$} B. ${ 6-2i\$} C. ${ 2i\$} D. { -2$}$ E. ${ 18\$}

by ADMIN 111 views

Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including algebra, geometry, and calculus. In this article, we will focus on simplifying a complex expression involving complex numbers. We will use the given expression {(8-5i) - (-2+3i)$}$ and simplify it step by step.

Understanding Complex Numbers

Before we dive into simplifying the expression, let's briefly review the concept of 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$}$. The real part of a complex number is {a$}$, and the imaginary part is {b$}$.

Simplifying the Expression

Now, let's simplify the given expression {(8-5i) - (-2+3i)$}$. To do this, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expressions inside the parentheses.
  2. Exponents: Evaluate any exponents (none in this case).
  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.

Step 1: Evaluate the Expressions Inside the Parentheses

The expression inside the parentheses is {(8-5i) - (-2+3i)$}$. To evaluate this expression, we need to follow the order of operations:

  1. Subtract βˆ’2+3i{-2+3i} from 8βˆ’5i{8-5i}.

Step 3: Subtract βˆ’2+3i{-2+3i} from 8βˆ’5i{8-5i}

To subtract βˆ’2+3i{-2+3i} from 8βˆ’5i{8-5i}, we need to subtract the real parts and the imaginary parts separately:

(8βˆ’5i)βˆ’(βˆ’2+3i)=(8βˆ’(βˆ’2))+(βˆ’5iβˆ’3i){(8-5i) - (-2+3i) = (8-(-2)) + (-5i-3i)}

Step 4: Simplify the Real and Imaginary Parts

Now, let's simplify the real and imaginary parts:

(8βˆ’(βˆ’2))+(βˆ’5iβˆ’3i)=(8+2)+(βˆ’5iβˆ’3i){(8-(-2)) + (-5i-3i) = (8+2) + (-5i-3i)}

Step 5: Combine Like Terms

Finally, let's combine like terms:

(8+2)+(βˆ’5iβˆ’3i)=10βˆ’8i{(8+2) + (-5i-3i) = 10 - 8i}

Conclusion

In conclusion, the simplified expression is ${10-8i\$}. This is the correct answer.

Final Answer

The final answer is ${10-8i\$}.

Discussion

The given expression {(8-5i) - (-2+3i)$}$ can be simplified by following the order of operations. We need to evaluate the expressions inside the parentheses, subtract the real parts and the imaginary parts separately, and finally combine like terms. The simplified expression is ${10-8i\$}.

Comparison with Other Options

Let's compare our answer with the other options:

  • A. ${10-8i\$}: This is the correct answer.
  • B. ${6-2i\$}: This is not the correct answer.
  • C. ${2i\$}: This is not the correct answer.
  • D. {-2$}$: This is not the correct answer.
  • E. ${18\$}: This is not the correct answer.

Conclusion

In conclusion, the simplified expression is ${10-8i\$}. This is the correct answer. We need to follow the order of operations to simplify complex expressions involving complex numbers.

Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including algebra, geometry, and calculus. In this article, we will focus on simplifying a complex expression involving complex numbers. We will use the given expression {(8-5i) - (-2+3i)$}$ and simplify it step by step.

Understanding Complex Numbers

Before we dive into simplifying the expression, let's briefly review the concept of 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$}$. The real part of a complex number is {a$}$, and the imaginary part is {b$}$.

Simplifying the Expression

Now, let's simplify the given expression {(8-5i) - (-2+3i)$}$. To do this, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expressions inside the parentheses.
  2. Exponents: Evaluate any exponents (none in this case).
  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.

Step 1: Evaluate the Expressions Inside the Parentheses

The expression inside the parentheses is {(8-5i) - (-2+3i)$}$. To evaluate this expression, we need to follow the order of operations:

  1. Subtract βˆ’2+3i{-2+3i} from 8βˆ’5i{8-5i}.

Step 3: Subtract βˆ’2+3i{-2+3i} from 8βˆ’5i{8-5i}

To subtract βˆ’2+3i{-2+3i} from 8βˆ’5i{8-5i}, we need to subtract the real parts and the imaginary parts separately:

(8βˆ’5i)βˆ’(βˆ’2+3i)=(8βˆ’(βˆ’2))+(βˆ’5iβˆ’3i){(8-5i) - (-2+3i) = (8-(-2)) + (-5i-3i)}

Step 4: Simplify the Real and Imaginary Parts

Now, let's simplify the real and imaginary parts:

(8βˆ’(βˆ’2))+(βˆ’5iβˆ’3i)=(8+2)+(βˆ’5iβˆ’3i){(8-(-2)) + (-5i-3i) = (8+2) + (-5i-3i)}

Step 5: Combine Like Terms

Finally, let's combine like terms:

(8+2)+(βˆ’5iβˆ’3i)=10βˆ’8i{(8+2) + (-5i-3i) = 10 - 8i}

Conclusion

In conclusion, the simplified expression is ${10-8i\$}. This is the correct answer.

Final Answer

The final answer is ${10-8i\$}.

Discussion

The given expression {(8-5i) - (-2+3i)$}$ can be simplified by following the order of operations. We need to evaluate the expressions inside the parentheses, subtract the real parts and the imaginary parts separately, and finally combine like terms. The simplified expression is ${10-8i\$}.

Comparison with Other Options

Let's compare our answer with the other options:

  • A. ${10-8i\$}: This is the correct answer.
  • B. ${6-2i\$}: This is not the correct answer.
  • C. ${2i\$}: This is not the correct answer.
  • D. {-2$}$: This is not the correct answer.
  • E. ${18\$}: This is not the correct answer.

Conclusion

In conclusion, the simplified expression is ${10-8i\$}. This is the correct answer. We need to follow the order of operations to simplify complex expressions involving complex numbers.

Q&A

Q: What is the correct answer for the expression {(8-5i) - (-2+3i)$}$?

A: The correct answer is ${10-8i\$}.

Q: How do we simplify complex expressions involving complex numbers?

A: We need to follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: What is the difference between the real and imaginary parts of a complex number?

A: The real part of a complex number is {a$}$, and the imaginary part is {b$}$.

Q: How do we subtract complex numbers?

A: We need to subtract the real parts and the imaginary parts separately.

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

A: The correct order of operations is: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: What is the final answer for the expression {(8-5i) - (-2+3i)$}$?

A: The final answer is ${10-8i\$}.

Q: How do we combine like terms in complex expressions?

A: We need to combine the real parts and the imaginary parts separately.

Q: What is the correct answer for the expression {(8-5i) - (-2+3i)$}$ compared to the other options?

A: The correct answer is ${10-8i\$}, and the other options are not correct.

Conclusion

In conclusion, the simplified expression is ${10-8i\$}. This is the correct answer. We need to follow the order of operations to simplify complex expressions involving complex numbers.