Select The Correct Answer.What Is $\sqrt{-16} + \sqrt{49}$ Written As A Complex Number In The Form $a + Bi$?A. $4 + 7i$ B. $7 - 4i$ C. $7 + 4i$ D. $-4 + 7i$

Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including engineering, physics, and computer science. In this article, we will explore how to solve complex numbers, specifically the expression $\sqrt{-16} + \sqrt{49}$, and write it in the form $a + bi$.

What are 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 $i^2 = -1$. Complex numbers can be represented graphically on a complex plane, with the real part $a$ on the x-axis and the imaginary part $b$ on the y-axis.

Solving the Expression

To solve the expression $\sqrt{-16} + \sqrt{49}$, we need to find the square roots of the negative and positive numbers separately.

Solving $\sqrt{-16}$

The square root of a negative number can be expressed as a complex number. To find the square root of $-16$, we can rewrite it as $-16 = 16 \cdot (-1)$. Since $i^2 = -1$, we can take the square root of $-16$ as follows:

$$\sqrt{-16} = \sqrt{16 \cdot (-1)} = \sqrt{16} \cdot \sqrt{-1} = 4i$$

Solving $\sqrt{49}$

The square root of a positive number is a real number. To find the square root of $49$, we can simply take the square root of $49$:

$$\sqrt{49} = 7$$

Combining the Results

Now that we have found the square roots of $-16$ and $49$, we can combine the results to get the final answer:

$$\sqrt{-16} + \sqrt{49} = 4i + 7 = 7 + 4i$$

Conclusion

In this article, we have solved the expression $\sqrt{-16} + \sqrt{49}$ and written it in the form $a + bi$. We have shown that the square root of a negative number can be expressed as a complex number, and we have combined the results to get the final answer. The correct answer is $7 + 4i$.

Answer Key

• A. $4 + 7i$
• B. $7 - 4i$
• C. $7 + 4i$
• D. $-4 + 7i$

The correct answer is C. $7 + 4i$.

Additional Resources

For more information on complex numbers, you can refer to the following resources:

• Khan Academy: Complex Numbers
• MIT OpenCourseWare: Complex Analysis
• Wolfram MathWorld: Complex Numbers

Frequently Asked Questions

• 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: How do I solve a complex number? A: To solve a complex number, you need to find the square roots of the negative and positive numbers separately, and then combine the results.
• Q: What is the imaginary unit? A: The imaginary unit is a number that satisfies the equation $i^2 = -1$. It is often represented by the letter $i$.
  Complex Numbers Q&A: Frequently Asked Questions =====================================================

Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including engineering, physics, and computer science. In this article, we will answer some of the most 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, which satisfies the equation $i^2 = -1$.

Q: How do I add complex numbers?

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

Q: How do I subtract complex numbers?

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

Q: How do I multiply complex numbers?

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

Q: How do I divide complex numbers?

A: To divide complex numbers, you need to multiply the numerator and the denominator by the conjugate of the denominator. For example, if you have two complex numbers $a + bi$ and $c + di$, the 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 of a complex number is used to simplify 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 $\sqrt{a^2 + b^2}$. The modulus of a complex number is a measure of its distance from the origin in the complex plane.

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 joining the origin to the point $(a, b)$ in the complex plane. The argument of a complex number is measured in radians.

Q: How do I convert a complex number from rectangular form to polar form?

A: To convert a complex number from rectangular form to polar form, you need to use the following formulas:

• $r = \sqrt{a^2 + b^2}$
• $\theta = \tan^{-1}\left(\frac{b}{a}\right)$

Q: How do I convert a complex number from polar form to rectangular form?

A: To convert a complex number from polar form to rectangular form, you need to use the following formulas:

• $a = r \cos \theta$
• $b = r \sin \theta$

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 covered how to convert complex numbers from rectangular form to polar form and vice versa.

Additional Resources

For more information on complex numbers, you can refer to the following resources:

• Khan Academy: Complex Numbers
• MIT OpenCourseWare: Complex Analysis
• Wolfram MathWorld: Complex Numbers

Frequently Asked Questions (FAQs)

• 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: How do I add complex numbers? A: To add complex numbers, you need to add the real parts and the imaginary parts separately.
• Q: How do I subtract complex numbers? A: To subtract complex numbers, you need to subtract the real parts and the imaginary parts separately.
• Q: How do I multiply complex numbers? A: To multiply complex numbers, you need to use the distributive property and the fact that $i^2 = -1$.
• Q: How do I divide complex numbers? A: To divide complex numbers, you need to multiply the numerator and the denominator by the conjugate of the denominator.

Glossary

• Complex number: A number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.
• Imaginary unit: A number that satisfies the equation $i^2 = -1$.
• Conjugate: The conjugate of a complex number $a + bi$ is $a - bi$.
• Modulus: The modulus of a complex number $a + bi$ is $\sqrt{a^2 + b^2}$.
• Argument: The argument of a complex number $a + bi$ is the angle between the positive real axis and the line segment joining the origin to the point $(a, b)$ in the complex plane.