Select The Correct Answer.What Is The Value Of The Expression Given Below?$ (8-3i) - (8-3i)(8+8i) $A. $ -80-43i $ B. $ -80+43i $ C. $ -96+37i $ D. $ -96-37i $
Introduction
Complex numbers are an essential part of mathematics, and they have numerous applications in various fields, including physics, engineering, and computer science. In this article, we will focus on simplifying complex expressions, specifically the expression given below: $ (8-3i) - (8-3i)(8+8i) $. We will break down the solution step by step, and by the end of this article, you will be able to simplify complex expressions with ease.
Understanding Complex Numbers
Before we dive into the solution, let's quickly review the basics 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 step by step.
Step 1: Multiply the Complex Numbers
To simplify the expression, we need to multiply the complex numbers $ (8-3i) $ and $ (8+8i) $. We can use the distributive property to multiply these numbers.
$ (8-3i)(8+8i) = 8(8+8i) - 3i(8+8i) $
$ = 64 + 64i - 24i - 24i^2 $
$ = 64 + 40i + 24 $
$ = 88 + 40i $
Step 2: Subtract the Result from the Original Expression
Now that we have multiplied the complex numbers, we can subtract the result from the original expression.
$ (8-3i) - (8-3i)(8+8i) = 8-3i - (88 + 40i) $
$ = 8 - 3i - 88 - 40i $
$ = -80 - 43i $
Conclusion
In this article, we simplified the complex expression $ (8-3i) - (8-3i)(8+8i) $ step by step. We multiplied the complex numbers using the distributive property and then subtracted the result from the original expression. The final answer is $ -80 - 43i $.
Answer Key
The correct answer is:
- A. $ -80-43i $
Why Choose This Answer?
This answer is the correct result of the expression $ (8-3i) - (8-3i)(8+8i) $. We obtained this result by multiplying the complex numbers and then subtracting the result from the original expression.
What's Next?
Simplifying complex expressions is an essential skill in mathematics, and it has numerous applications in various fields. In the next article, we will explore more complex expressions and learn how to simplify them using different techniques.
Additional Resources
If you want to learn more about complex numbers and simplifying expressions, here are some additional resources:
- Khan Academy: Complex Numbers
- MIT OpenCourseWare: Complex Analysis
- Wolfram MathWorld: Complex Numbers
Final Thoughts
Introduction
In our previous article, we simplified the complex expression $ (8-3i) - (8-3i)(8+8i) $ step by step. We received many questions from readers who wanted to learn more about simplifying complex expressions. In this article, we will answer some of the most frequently asked questions about simplifying complex expressions.
Q: What is the difference between a real number and a complex number?
A: A real number is a number that can be expressed without any imaginary part, such as 3, 4, or 5. A complex number, on the other hand, 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 $.
Q: How do I multiply complex numbers?
A: To multiply complex numbers, you can use the distributive property. For example, to multiply $ (8-3i) $ and $ (8+8i) $, you would multiply each term in the first expression by each term in the second expression.
$ (8-3i)(8+8i) = 8(8+8i) - 3i(8+8i) $
$ = 64 + 64i - 24i - 24i^2 $
$ = 64 + 40i + 24 $
$ = 88 + 40i $
Q: How do I simplify complex expressions?
A: To simplify complex expressions, you can use the following steps:
- Multiply complex numbers using the distributive property.
- Combine like terms.
- Simplify the expression by combining real and imaginary parts.
Q: What is the imaginary unit, and how is it used in complex numbers?
A: The imaginary unit, denoted by $ i $, is a mathematical concept that satisfies the equation $ i^2 = -1 $. It is used to represent the imaginary part of a complex number.
Q: Can you provide more examples of simplifying complex expressions?
A: Here are a few more examples of simplifying complex expressions:
- $ (3+4i) - (3+4i)(2-2i) = 3+4i - (6-6i+8i-8i^2) = 3+4i - (6-2i+8) = -1+2i $
- $ (2-3i) + (2-3i)(4+4i) = 2-3i + (8+8i-12i-12i^2) = 2-3i + (8-4i+12) = 14-4i $
- $ (5+2i) - (5+2i)(3-3i) = 5+2i - (15-15i+6i-6i^2) = 5+2i - (15-9i+6) = -10+9i $
Conclusion
In this article, we answered some of the most frequently asked questions about simplifying complex expressions. We covered topics such as the difference between real and complex numbers, multiplying complex numbers, simplifying complex expressions, and the imaginary unit. We hope that this article has provided you with a better understanding of simplifying complex expressions and has given you the confidence to tackle more complex problems.
Additional Resources
If you want to learn more about complex numbers and simplifying expressions, here are some additional resources:
- Khan Academy: Complex Numbers
- MIT OpenCourseWare: Complex Analysis
- Wolfram MathWorld: Complex Numbers
Final Thoughts
Simplifying complex expressions is a crucial skill in mathematics, and it requires practice and patience. We hope that this article has helped you understand how to simplify complex expressions and has provided you with the confidence to tackle more complex problems. Remember to practice regularly and seek help when you need it. With time and practice, you will become proficient in simplifying complex expressions and will be able to tackle even the most challenging problems.