Perform The Indicated Operation And Simplify. Express The Answer As A Complex Number.\[$(8+5i)(2-i) = \square\$\]

Introduction

Complex numbers are mathematical expressions that consist of a real part and an imaginary part. They are used to represent quantities that have both magnitude and direction. In this article, we will explore how to perform operations with complex numbers, specifically multiplication and simplification. We will use the given expression $(8+5i)(2-i)$ as an example to demonstrate the steps involved.

Understanding 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 the part that is not multiplied by $i$, while the imaginary part is the part that is multiplied by $i$.

Multiplying Complex Numbers

To multiply two complex numbers, we use the distributive property, which states that $(a+b)(c+d) = ac + ad + bc + bd$. We can apply this property to multiply two complex numbers by multiplying each term in the first number by each term in the second number.

Step 1: Multiply the Real Parts

The first step in multiplying complex numbers is to multiply the real parts. In the given expression $(8+5i)(2-i)$, the real parts are $8$ and $2$. We multiply these two numbers together to get:

$$8 \times 2 = 16$$

Step 2: Multiply the Imaginary Parts

The next step is to multiply the imaginary parts. In the given expression, the imaginary parts are $5i$ and $-i$. We multiply these two numbers together to get:

$$5i \times -i = -5i^2$$

Since $i^2 = -1$, we can substitute this value into the expression to get:

$$-5i^2 = -5(-1) = 5$$

Step 3: Multiply the Cross Terms

The final step is to multiply the cross terms. In the given expression, the cross terms are $8i$ and $-2i$. We multiply these two numbers together to get:

$$8i \times -2i = -16i^2$$

Since $i^2 = -1$, we can substitute this value into the expression to get:

$$-16i^2 = -16(-1) = 16$$

Step 4: Combine the Terms

Now that we have multiplied all the terms, we can combine them to get the final result. We add the real parts together to get:

$$16 + 5 + 16 = 37$$

We add the imaginary parts together to get:

$$-5i + 16i = 11i$$

Therefore, the final result is:

$$(8+5i)(2-i) = 37 + 11i$$

Conclusion

In this article, we have demonstrated how to perform operations with complex numbers, specifically multiplication and simplification. We used the given expression $(8+5i)(2-i)$ as an example to show the steps involved. By following these steps, we can simplify complex expressions and express the answer as a complex number.

Key Takeaways

• Complex numbers are mathematical expressions that consist of a real part and an imaginary part.
• To multiply complex numbers, we use the distributive property and multiply each term in the first number by each term in the second number.
• We can simplify complex expressions by combining the real and imaginary parts.
• The final result is a complex number in the form $a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Further Reading

If you want to learn more about complex numbers and operations, here are some additional resources:

• Complex Numbers on Wikipedia
• Complex Numbers on Math Is Fun
• Complex Numbers on Khan Academy

Practice Problems

Try these practice problems to test your understanding of complex numbers and operations:

• $(3+4i)(2+5i) = \square$
• $(6-2i)(4+3i) = \square$
• $(1+2i)(3-4i) = \square$

Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and computer science. However, complex numbers can be challenging to understand, especially for beginners. In this article, we will address some of the most frequently asked questions about complex numbers, providing clear and concise answers to help you better understand this complex topic.

Q: What is a complex number?

A: A complex number is a mathematical expression that consists of a real part and an imaginary part. It is typically represented 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 $i$?

A: The imaginary unit $i$ is a mathematical constant that is defined as the square root of $-1$. It is denoted by the letter $i$ and is used to represent the imaginary part of a complex number.

Q: How do I add complex numbers?

A: To add complex numbers, you simply add the real parts together and the imaginary parts together. For example, if you have two complex numbers $a+bi$ and $c+di$, their sum is $(a+c)+(b+d)i$.

Q: How do I subtract complex numbers?

A: To subtract complex numbers, you simply subtract the real parts together and the imaginary parts together. For example, if you have two complex numbers $a+bi$ and $c+di$, their difference is $(a-c)+(b-d)i$.

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you use the distributive property and multiply each term in the first number by each term in the second number. For example, if you have two complex numbers $a+bi$ and $c+di$, their product is $(ac-bd)+(ad+bc)i$.

Q: How do I divide complex numbers?

A: To divide complex numbers, you multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number $a+bi$ is $a-bi$. For example, if you have two complex numbers $a+bi$ and $c+di$, their quotient is $\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}$.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number $a+bi$ is $a-bi$. The conjugate is used to simplify complex expressions and to divide complex numbers.

Q: What is the modulus of a complex number?

A: The modulus of a complex number $a+bi$ is the distance from the origin to the point $(a,b)$ in the complex plane. It is denoted by $|a+bi|$ and is calculated as $\sqrt{a^2+b^2}$.

Q: What is the argument of a complex number?

A: The argument of a complex number $a+bi$ is the angle between the positive real axis and the line segment connecting the origin to the point $(a,b)$ in the complex plane. It is denoted by $\arg(a+bi)$ and is calculated as $\tan^{-1}\left(\frac{b}{a}\right)$.

Q: What are some common applications of complex numbers?

A: Complex numbers have numerous applications in various fields, including:

• Physics: Complex numbers are used to describe the behavior of electrical circuits, mechanical systems, and other physical systems.
• Engineering: Complex numbers are used to design and analyze electrical circuits, mechanical systems, and other engineering systems.
• Computer Science: Complex numbers are used in algorithms for solving linear systems, finding eigenvalues, and other computational tasks.
• Signal Processing: Complex numbers are used to analyze and process signals in various fields, including audio, image, and video processing.

Conclusion

In this article, we have addressed some of the most frequently asked questions about complex numbers, providing clear and concise answers to help you better understand this complex topic. We hope that this article has been helpful in your journey to learn about complex numbers and their applications.

Key Takeaways

• Complex numbers are mathematical expressions that consist of a real part and an imaginary part.
• The imaginary unit $i$ is a mathematical constant that is defined as the square root of $-1$.
• To add complex numbers, you simply add the real parts together and the imaginary parts together.
• To subtract complex numbers, you simply subtract the real parts together and the imaginary parts together.
• To multiply complex numbers, you use the distributive property and multiply each term in the first number by each term in the second number.
• To divide complex numbers, you multiply the numerator and denominator by the conjugate of the denominator.
• The conjugate of a complex number is used to simplify complex expressions and to divide complex numbers.
• The modulus of a complex number is the distance from the origin to the point $(a,b)$ in the complex plane.
• The argument of a complex number is the angle between the positive real axis and the line segment connecting the origin to the point $(a,b)$ in the complex plane.

Further Reading

If you want to learn more about complex numbers and their applications, here are some additional resources:

• Complex Numbers on Wikipedia
• Complex Numbers on Math Is Fun
• Complex Numbers on Khan Academy

Practice Problems

Try these practice problems to test your understanding of complex numbers and their operations:

• $(3+4i)+(2+5i) = \square$
• $(6-2i)-(4+3i) = \square$
• $(1+2i)(3-4i) = \square$
• $\frac{(1+2i)}{(3+4i)} = \square$

Remember to follow the steps outlined in this article to simplify the expressions and express the answer as a complex number.