Find The Product Of These Complex Numbers:\[$(5+6i)(3-3i) =\$\]A. \[$-3 + 3i\$\]B. \[$33 + 3i\$\]C. \[$15 - 18i\$\]D. \[$15 + 181\$\]
Introduction
Complex numbers are an extension of the real number system, where each number is represented in the form of a + bi, where 'a' is the real part and 'bi' is the imaginary part. In this article, we will focus on finding the product of two complex numbers, (5+6i) and (3-3i). We will use the distributive property of multiplication to simplify the expression and find the final product.
Understanding Complex Number Multiplication
To multiply two complex numbers, we can use the distributive property of multiplication. This property states that for any complex numbers a + bi and c + di, the product is given by:
(a + bi)(c + di) = ac + adi + bci + bdi^2
Since i^2 = -1, we can simplify the expression as:
(a + bi)(c + di) = ac - bd + (ad + bc)i
Applying the Distributive Property
Now, let's apply the distributive property to find the product of (5+6i) and (3-3i).
(5+6i)(3-3i) = 5(3-3i) + 6i(3-3i)
Using the distributive property, we can expand the expression as:
5(3-3i) = 15 - 15i 6i(3-3i) = 18i - 18i^2
Since i^2 = -1, we can simplify the expression as:
18i - 18i^2 = 18i + 18
Now, let's combine the two expressions:
15 - 15i + 18i + 18
Simplifying the Expression
To simplify the expression, we can combine like terms:
15 + 18 = 33 -15i + 18i = 3i
Therefore, the product of (5+6i) and (3-3i) is:
33 + 3i
Conclusion
In this article, we used the distributive property of multiplication to find the product of two complex numbers, (5+6i) and (3-3i). We simplified the expression by combining like terms and found the final product to be 33 + 3i.
Answer
The correct answer is:
B. 33 + 3i
Practice Problems
- Find the product of (2+3i) and (4-2i).
- Find the product of (1-2i) and (3+4i).
- Find the product of (5-6i) and (2+3i).
Solutions
-
(2+3i)(4-2i) = 8 - 4i + 12i - 6i^2 = 8 + 8i + 6 = 14 + 8i
-
(1-2i)(3+4i) = 3 + 4i - 6i - 8i^2 = 3 - 2i + 8 = 11 - 2i
-
(5-6i)(2+3i) = 10 + 15i - 12i - 18i^2 = 10 + 3i + 18 = 28 + 3i
Final Thoughts
Introduction
In our previous article, we discussed how to multiply two complex numbers using the distributive property of multiplication. We also provided practice problems and solutions to help readers understand the concept better. In this article, we will answer some frequently asked questions about complex number multiplication.
Q&A
Q: What is the formula for multiplying two complex numbers?
A: The formula for multiplying two complex numbers is:
(a + bi)(c + di) = ac - bd + (ad + bc)i
Q: How do I apply the distributive property to multiply two complex numbers?
A: To apply the distributive property, you need to multiply each term in the first complex number by each term in the second complex number. For example, if you want to multiply (5+6i) and (3-3i), you would multiply 5 by 3, 5 by -3i, 6i by 3, and 6i by -3i.
Q: What is the difference between multiplying two complex numbers and multiplying two real numbers?
A: The main difference between multiplying two complex numbers and multiplying two real numbers is that complex numbers have an imaginary part, which is denoted by 'i'. When you multiply two complex numbers, you need to take into account the imaginary part and use the distributive property to simplify the expression.
Q: Can I use the commutative property to multiply two complex numbers?
A: Yes, you can use the commutative property to multiply two complex numbers. The commutative property states that the order of the factors does not change the result. For example, (a + bi)(c + di) = (c + di)(a + bi).
Q: How do I simplify the expression after multiplying two complex numbers?
A: To simplify the expression, you need to combine like terms. For example, if you have the expression 15 - 15i + 18i + 18, you can combine the like terms to get 33 + 3i.
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 use the distributive property
- Not taking into account the imaginary part
- Not combining like terms
- Not using the correct formula
Q: Can I use a calculator to multiply complex numbers?
A: Yes, you can use a calculator to multiply complex numbers. However, it's always a good idea to understand the concept and be able to multiply complex numbers by hand.
Q: How do I apply complex number multiplication in real-life situations?
A: Complex number multiplication has many real-life 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 represent the behavior of control systems and analyze their stability.
Conclusion
In this article, we answered some frequently asked questions about complex number multiplication. We covered topics such as the formula for multiplying two complex numbers, applying the distributive property, and simplifying the expression after multiplication. We also discussed common mistakes to avoid and real-life applications of complex number multiplication.
Practice Problems
- Multiply (2+3i) and (4-2i).
- Multiply (1-2i) and (3+4i).
- Multiply (5-6i) and (2+3i).
Solutions
-
(2+3i)(4-2i) = 8 - 4i + 12i - 6i^2 = 8 + 8i + 6 = 14 + 8i
-
(1-2i)(3+4i) = 3 + 4i - 6i - 8i^2 = 3 - 2i + 8 = 11 - 2i
-
(5-6i)(2+3i) = 10 + 15i - 12i - 18i^2 = 10 + 3i + 18 = 28 + 3i
Final Thoughts
In this article, we provided a Q&A guide to complex number multiplication. We covered topics such as the formula for multiplying two complex numbers, applying the distributive property, and simplifying the expression after multiplication. We also discussed common mistakes to avoid and real-life applications of complex number multiplication.