What Is The Sum Of − 2 \sqrt{-2} − 2 ​ And − 18 \sqrt{-18} − 18 ​ ?A. 4 2 4 \sqrt{2} 4 2 ​ B. 4 I 2 4 I \sqrt{2} 4 I 2 ​ C. 5 2 5 \sqrt{2} 5 2 ​ D. 5 I 2 5 I \sqrt{2} 5 I 2 ​

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Introduction

When dealing with square roots of negative numbers, we are essentially working with complex numbers. In mathematics, complex numbers are numbers 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 explore the sum of $\sqrt{-2}$ and $\sqrt{-18}$, and determine which of the given options is the correct answer.

Understanding Square Roots of Negative Numbers

To begin with, let's understand what the square root of a negative number means. When we take the square root of a negative number, we are essentially looking for a number that, when multiplied by itself, gives us a negative result. However, since the square of any real number is always non-negative, we cannot find a real number that satisfies this condition. Therefore, we need to extend the real number system to include complex numbers.

Expressing Square Roots of Negative Numbers

We can express the square root of a negative number in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit. For example, the square root of $-2$ can be expressed as $\sqrt{-2} = \sqrt{2}i$, since $(\sqrt{2}i)^2 = -2$. Similarly, the square root of $-18$ can be expressed as $\sqrt{-18} = \sqrt{18}i = 3\sqrt{2}i$, since $(3\sqrt{2}i)^2 = -18$.

Finding the Sum

Now that we have expressed $\sqrt{-2}$ and $\sqrt{-18}$ in terms of complex numbers, we can find their sum. The sum of $\sqrt{-2}$ and $\sqrt{-18}$ is given by:

$$\sqrt{-2} + \sqrt{-18} = \sqrt{2}i + 3\sqrt{2}i = 4\sqrt{2}i$$

Conclusion

In conclusion, the sum of $\sqrt{-2}$ and $\sqrt{-18}$ is $4\sqrt{2}i$. This is the correct answer, and it can be verified by substituting the values into the equation. Therefore, the correct option is:

A. $4 \sqrt{2}$

However, this option is not correct, since the sum is actually $4\sqrt{2}i$, not $4\sqrt{2}$. The correct option is actually:

B. $4 i \sqrt{2}$

This option correctly represents the sum of $\sqrt{-2}$ and $\sqrt{-18}$ as $4\sqrt{2}i$.

Final Answer

The final answer is $\boxed{B. $4 i \sqrt{2}$}$.

Frequently Asked Questions

Q: What is the square root of a negative number?

A: The square root of a negative number is a complex number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: How do you express the square root of a negative number?

A: You can express the square root of a negative number in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: What is the sum of $\sqrt{-2}$ and $\sqrt{-18}$?

A: The sum of $\sqrt{-2}$ and $\sqrt{-18}$ is $4\sqrt{2}i$.

References

• [1] "Complex Numbers" by Math Open Reference. Retrieved from https://www.mathopenref.com/complexnumbers.html
• [2] "Square Root of a Negative Number" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/SquareRootofaNegativeNumber.html

Related Articles

• [1] "Introduction to Complex Numbers"
• [2] "Properties of Complex Numbers"
• [3] "Operations with Complex Numbers"
  

Introduction

Complex numbers are an extension of the real number system, which includes numbers that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit. In this article, we will answer some frequently asked questions about complex numbers.

Q&A

Q: What is a complex number?

A: 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.

Q: What is the imaginary unit?

A: The imaginary unit, denoted by $i$, is a number that satisfies the equation $i^2 = -1$. It is used to extend the real number system to include complex numbers.

Q: How do you add complex numbers?

A: To add complex numbers, you add the real parts and the imaginary parts separately. For example, $(a + bi) + (c + di) = (a + c) + (b + d)i$.

Q: How do you subtract complex numbers?

A: To subtract complex numbers, you subtract the real parts and the imaginary parts separately. For example, $(a + bi) - (c + di) = (a - c) + (b - d)i$.

Q: How do you multiply complex numbers?

A: To multiply complex numbers, you use the distributive property and the fact that $i^2 = -1$. For example, $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$.

Q: How do you divide complex numbers?

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

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number $a + bi$ is $a - bi$.

Q: How do you find the magnitude of a complex number?

A: The magnitude of a complex number $a + bi$ is given by $\sqrt{a^2 + b^2}$.

Q: How do you find the argument of a complex number?

A: The argument of a complex number $a + bi$ is given by $\tan^{-1}\left(\frac{b}{a}\right)$.

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

A: 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 $\theta = \tan^{-1}\left(\frac{b}{a}\right)$.

Conclusion

In conclusion, complex numbers are an extension of the real number system, which includes numbers that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit. We have answered some frequently asked questions about complex numbers, including how to add, subtract, multiply, and divide complex numbers, as well as how to find the magnitude and argument of a complex number.

Final Answer

The final answer is $\boxed{B. $4 i \sqrt{2}$}$.

Frequently Asked Questions

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

A: 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. A real number is a number that can be expressed in the form $a$, where $a$ is a real number.

Q: How do you convert a complex number to a real number?

A: You can convert a complex number to a real number by setting the imaginary part to zero. For example, $a + bi$ becomes $a$.

Q: How do you convert a real number to a complex number?

A: You can convert a real number to a complex number by setting the imaginary part to zero. For example, $a$ becomes $a + 0i$.

References

• [1] "Complex Numbers" by Math Open Reference. Retrieved from https://www.mathopenref.com/complexnumbers.html
• [2] "Square Root of a Negative Number" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/SquareRootofaNegativeNumber.html

Related Articles

• [1] "Introduction to Complex Numbers"
• [2] "Properties of Complex Numbers"
• [3] "Operations with Complex Numbers"