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Mar 13, 2025 by ADMIN 230 views

Simplify $-6i(5+3i)$

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Introduction

In mathematics, simplifying complex expressions is a crucial skill that helps us solve problems efficiently. One of the most common operations in algebra is multiplying complex numbers. In this article, we will focus on simplifying the expression $-6i(5+3i)$ , which involves multiplying a complex number by a scalar and another complex number.

Understanding Complex Numbers

Before we dive into simplifying the expression, let's quickly review what complex numbers are. 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 $b$ .

Multiplying Complex Numbers

To simplify the expression $-6i(5+3i)$ , we need to multiply the complex number $-6i$ by the complex number $5+3i$ . When multiplying complex numbers, we follow the distributive property, which states that for any complex numbers $a+bi$ and $c+di$ , the product is given by:

$(a+bi)(c+di) = (ac-bd) + (ad+bc)i$

Applying the Distributive Property

Now, let's apply the distributive property to simplify the expression $-6i(5+3i)$ . We can start by multiplying the scalar $-6i$ by the complex number $5+3i$ :

$-6i(5+3i) = (-6i)(5) + (-6i)(3i)$

Simplifying the Expression

Next, we simplify each term separately. The first term is $(-6i)(5)$ , which is equal to $-30i$ . The second term is $(-6i)(3i)$ , which is equal to $-18i^2$ . Since $i^2 = -1$ , we can substitute this value into the expression:

$-18i^2 = -18(-1) = 18$

Combining the Terms

Now, we combine the two terms to get the final simplified expression:

$-30i + 18$

Writing the Answer in Standard Form

The final answer should be written in standard form, which is $a+bi$ . In this case, the real part is $18$ , and the imaginary part is $-30$ . Therefore, the final answer is:

$18-30i$

Conclusion

In this article, we simplified the expression $-6i(5+3i)$ by applying the distributive property and simplifying each term separately. We also reviewed the basics of complex numbers and multiplication. By following these steps, we can simplify complex expressions and solve problems efficiently.

Frequently Asked Questions

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, whereas a complex number is a number that has both real and imaginary parts.

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you follow the distributive property, which states that for any complex numbers $a+bi$ and $c+di$ , the product is given by:

$(a+bi)(c+di) = (ac-bd) + (ad+bc)i$

Q: What is the standard form of a complex number?

A: The standard form of a complex number is $a+bi$ , where $a$ is the real part and $b$ is the imaginary part.

Additional Resources

For more information on complex numbers and multiplication, check out the following resources:

Khan Academy: Complex Numbers
Mathway: Complex Numbers
Wolfram MathWorld: Complex Numbers

Final Answer

The final answer is: $\boxed{18-30i}$

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Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and computer science. In this article, we will answer some of the most frequently asked questions about complex numbers.

Q: What is a complex number?

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 $i^2 = -1$ .

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

A real number is a number that can be expressed without any imaginary part, whereas a complex number is a number that has both real and imaginary parts.

Q: How do I add complex numbers?

To add complex numbers, you simply add the real parts and the imaginary parts separately. For example, if you have two complex numbers $a+bi$ and $c+di$ , their sum is given by:

$(a+bi) + (c+di) = (a+c) + (b+d)i$

Q: How do I subtract complex numbers?

To subtract complex numbers, you simply subtract the real parts and the imaginary parts separately. For example, if you have two complex numbers $a+bi$ and $c+di$ , their difference is given by:

$(a+bi) - (c+di) = (a-c) + (b-d)i$

Q: How do I multiply complex numbers?

To multiply complex numbers, you follow the distributive property, which states that for any complex numbers $a+bi$ and $c+di$ , the product is given by:

$(a+bi)(c+di) = (ac-bd) + (ad+bc)i$

Q: How do I divide complex numbers?

To divide complex numbers, you multiply the numerator and the denominator by the conjugate of the denominator. For example, if you have two complex numbers $a+bi$ and $c+di$ , their quotient is given by:

$\frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{(c+di)(c-di)}$

Q: What is the conjugate of a complex number?

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

Q: What is the modulus of a complex number?

The modulus of a complex number $a+bi$ is given by $\sqrt{a^2+b^2}$ . The modulus of a complex number is used to measure its distance from the origin in the complex plane.

Q: What is the argument of a complex number?

The argument of a complex number $a+bi$ is given by $\tan^{-1}\left(\frac{b}{a}\right)$ . The argument of a complex number is used to determine its position in the complex plane.

Q: What is the polar form of a complex number?

The polar form of a complex number $a+bi$ is given by $r(\cos\theta + i\sin\theta)$ , where $r = \sqrt{a^2+b^2}$ and $\theta = \tan^{-1}\left(\frac{b}{a}\right)$ .

Q: What is the exponential form of a complex number?

The exponential form of a complex number $a+bi$ is given by $re^{i\theta}$ , where $r = \sqrt{a^2+b^2}$ and $\theta = \tan^{-1}\left(\frac{b}{a}\right)$ .

Q: What are some common applications of complex numbers?

Complex numbers have numerous applications in various fields, including:

Physics: Complex numbers are used to describe the behavior of electrical circuits, the motion of objects, and the properties of materials.
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 performing other numerical computations.
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 answered some of the most frequently asked questions about complex numbers. We have covered topics such as adding, subtracting, multiplying, and dividing complex numbers, as well as the conjugate, modulus, and argument of a complex number. We have also discussed the polar and exponential forms of a complex number and some common applications of complex numbers.