Express The Following In The Simplest $a + B I$ Form:$\sqrt{9} + \sqrt{-36}$A. $-9 I$ B. $3 - 6 I$ C. $9 I$ D. $3 + 6 I$

Introduction

In mathematics, complex numbers are a fundamental concept that extends the real number system to include numbers with both real and imaginary parts. The simplest form of a complex number is represented as $a + bi$, where $a$ is the real part and $b$ is the imaginary part. In this article, we will explore how to express the given expression $\sqrt{9} + \sqrt{-36}$ in its simplest form.

Understanding Complex Numbers

Before we dive into the problem, let's briefly review the concept of 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 denoted by $a$, and the imaginary part is denoted by $b$.

Simplifying the Given Expression

Now, let's simplify the given expression $\sqrt{9} + \sqrt{-36}$. We can start by evaluating the square roots of the individual numbers.

• $\sqrt{9} = 3$, since $3^2 = 9$
• $\sqrt{-36} = \sqrt{-1} \cdot \sqrt{36} = 6i$, since $i^2 = -1$ and $\sqrt{36} = 6$

Now, we can substitute these values back into the original expression:

$\sqrt{9} + \sqrt{-36} = 3 + 6i$

Expressing the Result in Simplest Form

The result we obtained is already in its simplest form, $3 + 6i$. This is because we have expressed the complex number in the form $a + bi$, where $a = 3$ and $b = 6$.

Conclusion

In conclusion, we have successfully expressed the given expression $\sqrt{9} + \sqrt{-36}$ in its simplest form, $3 + 6i$. This result demonstrates the importance of understanding complex numbers and their representation in the form $a + bi$.

Answer

The correct answer is D. $3 + 6 i$.

Additional Examples

To further reinforce your understanding of complex numbers, let's consider a few additional examples:

• $\sqrt{16} + \sqrt{-25} = 4 - 5i$
• $\sqrt{9} + \sqrt{-49} = 3 - 7i$
• $\sqrt{25} + \sqrt{-36} = 5 + 6i$

These examples illustrate the process of simplifying complex numbers and expressing them in their simplest form.

Tips and Tricks

When working with complex numbers, it's essential to remember the following tips and tricks:

• Always simplify the square roots of individual numbers before combining them.
• Use the imaginary unit $i$ to represent the imaginary part of a complex number.
• Express complex numbers in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

Frequently Asked Questions

In this article, we'll address some of the most common questions related to complex numbers. Whether you're a student, a teacher, or simply someone interested in mathematics, this Q&A section will provide you with a deeper understanding of complex numbers.

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: What is the imaginary unit $i$?

A: The imaginary unit $i$ is a mathematical concept that represents the square root of $-1$. It's denoted by the letter $i$ and is used to extend the real number system to include complex numbers.

Q: How do I simplify complex numbers?

A: To simplify complex numbers, you can follow these steps:

1. Simplify the square roots of individual numbers.
2. Combine the real and imaginary parts.
3. Express the result in the form $a + bi$.

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, such as $3$ or $-4$. A complex number, on the other hand, is a number that has both real and imaginary parts, such as $3 + 4i$ or $-2 - 5i$.

Q: Can complex numbers be used in real-world applications?

A: Yes, complex numbers have numerous real-world applications, including:

• Electrical engineering: Complex numbers are used to analyze and design electrical circuits.
• Signal processing: Complex numbers are used to filter and analyze signals.
• Quantum mechanics: Complex numbers are used to describe the behavior of particles at the quantum level.

Q: How do I add and subtract complex numbers?

A: To add and subtract complex numbers, you can follow these rules:

• Add or subtract the real parts separately.
• Add or subtract the imaginary parts separately.
• Combine the results to form a new complex number.

Q: Can complex numbers be multiplied and divided?

A: Yes, complex numbers can be multiplied and divided using the following rules:

• Multiply the real parts together and the imaginary parts together.
• Add the products of the real and imaginary parts to form a new complex number.
• Divide the complex numbers by multiplying the numerator and denominator by the conjugate of the denominator.

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 numbers and to divide complex numbers.

Q: Can complex numbers be used to solve equations?

A: Yes, complex numbers can be used to solve equations, including quadratic equations and polynomial equations.

Q: What are some common mistakes to avoid when working with complex numbers?

A: Some common mistakes to avoid when working with complex numbers include:

• Forgetting to simplify the square roots of individual numbers.
• Not combining the real and imaginary parts correctly.
• Not using the imaginary unit $i$ correctly.

By following these tips and avoiding common mistakes, you'll be well on your way to mastering the art of working with complex numbers.