Multiply The Expression: \[$-3i(-6+6i)\$\].Write Your Answer As A Complex Number In Standard Form.
Introduction
In mathematics, complex numbers are a fundamental concept that plays a crucial role in various branches of mathematics, including algebra, geometry, and calculus. 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. In this article, we will focus on multiplying complex numbers, specifically the expression -3i(-6+6i).
Understanding Complex Numbers
Before we dive into multiplying complex numbers, 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. The real part of a complex number is denoted by a, and the imaginary part is denoted by bi.
Multiplying Complex Numbers
To multiply complex numbers, we can use the distributive property of multiplication over addition. This means that we can multiply each term in the first complex number by each term in the second complex number.
Step 1: Multiply the Real Parts
The first step in multiplying complex numbers is to multiply the real parts. In this case, we have -3i(-6+6i). To multiply the real parts, we can use the distributive property of multiplication over addition.
-3i(-6) = 18i
-3i(6i) = -18i^2
Step 2: Simplify the Imaginary Parts
Now that we have multiplied the real parts, we need to simplify the imaginary parts. Recall that i^2 = -1. Therefore, we can simplify the expression -18i^2 as follows:
-18i^2 = -18(-1) = 18
Step 3: Combine the Real and Imaginary Parts
Now that we have simplified the imaginary parts, we can combine the real and imaginary parts to get the final result.
18i + 18
Conclusion
In this article, we have multiplied the complex number -3i(-6+6i) using the distributive property of multiplication over addition. We have also simplified the imaginary parts using the fact that i^2 = -1. The final result is 18 + 18i.
Standard Form
The standard form of a complex number is a + bi, where a and b are real numbers, and i is the imaginary unit. In this case, the standard form of the complex number 18 + 18i is:
18 + 18i
Key Takeaways
- To multiply complex numbers, we can use the distributive property of multiplication over addition.
- We can simplify the imaginary parts using the fact that i^2 = -1.
- The standard form of a complex number is a + bi, where a and b are real numbers, and i is the imaginary unit.
Practice Problems
- Multiply the complex number 2i(3-4i).
- Multiply the complex number -5i(2+3i).
- Multiply the complex number 4i(1-2i).
Solutions
- 2i(3-4i) = 6i - 8i^2 = 6i + 8 = 8 + 6i.
- -5i(2+3i) = -10i - 15i^2 = -10i + 15 = 15 - 10i.
- 4i(1-2i) = 4i - 8i^2 = 4i + 8 = 8 + 4i.
Conclusion
Introduction
In our previous article, we discussed multiplying complex numbers using the distributive property of multiplication over addition. We also simplified the imaginary parts using the fact that i^2 = -1. In this article, we will provide a Q&A guide to help you understand and practice multiplying complex numbers.
Q: What is the distributive property of multiplication over addition?
A: The distributive property of multiplication over addition is a mathematical property that states that multiplication can be distributed over addition. In other words, we can multiply each term in the first complex number by each term in the second complex number.
Q: How do I multiply complex numbers?
A: To multiply complex numbers, we can use the distributive property of multiplication over addition. We can multiply each term in the first complex number by each term in the second complex number.
Q: What is the standard form of a complex number?
A: The standard form of a complex number is a + bi, where a and b are real numbers, and i is the imaginary unit.
Q: How do I simplify the imaginary parts of a complex number?
A: To simplify the imaginary parts of a complex number, we can use the fact that i^2 = -1. We can substitute i^2 with -1 and simplify the expression.
Q: What are some common mistakes to avoid when multiplying complex numbers?
A: Some common mistakes to avoid when multiplying complex numbers include:
- Forgetting to distribute the multiplication over addition
- Not simplifying the imaginary parts correctly
- Not using the correct order of operations
Q: How can I practice multiplying complex numbers?
A: You can practice multiplying complex numbers by working through practice problems and exercises. You can also use online resources and tools to help you practice and learn.
Q: What are some real-world applications of multiplying complex numbers?
A: Multiplying complex numbers has 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.
Q: Can I use a calculator to multiply complex numbers?
A: Yes, you can use a calculator to multiply complex numbers. However, it's also important to understand the underlying mathematical concepts and be able to perform the calculations by hand.
Q: How can I use multiplying complex numbers in my daily life?
A: Multiplying complex numbers can be used in many real-world applications, including:
- Analyzing AC circuits and designing electrical systems
- Analyzing signals and designing signal processing systems
- Analyzing and designing control systems
Conclusion
In this article, we have provided a Q&A guide to help you understand and practice multiplying complex numbers. We have also discussed the distributive property of multiplication over addition, the standard form of a complex number, and how to simplify the imaginary parts. We have also provided some common mistakes to avoid and some real-world applications of multiplying complex numbers.