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

Introduction

Complex numbers are mathematical expressions that consist of a real part and an imaginary part. They are often represented in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part. In this article, we will explore how to express complex numbers in their simplest form, using the given expression $\sqrt{9} + \sqrt{-36}$ as an example.

Understanding Complex Numbers

Before we dive into the example, 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 the part that is not multiplied by $i$, while the imaginary part is the part that is multiplied by $i$.

Simplifying the Given Expression

Now, let's simplify the given expression $\sqrt{9} + \sqrt{-36}$. To do this, we need to first simplify the square roots.

Simplifying the Square Roots

The square root of 9 can be simplified as follows:

$$\sqrt{9} = \sqrt{3^2} = 3$$

The square root of -36 can be simplified as follows:

$$\sqrt{-36} = \sqrt{-6^2} = 6i$$

Combining the Simplified Expressions

Now that we have simplified the square roots, we can combine the expressions to get the final result:

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

Conclusion

In this article, we have seen how to express complex numbers in their simplest form using the given expression $\sqrt{9} + \sqrt{-36}$ as an example. We simplified the square roots and combined the expressions to get the final result, which is $3 + 6i$. This result is in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

Answer

The correct answer is:

• D. $3 + 6i$

Why is this the correct answer?

This is the correct answer because we simplified the square roots and combined the expressions to get the final result, which is $3 + 6i$. This result is in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

Tips and Tricks

Here are some tips and tricks to help you simplify complex numbers:

• Use the properties of square roots: When simplifying square roots, remember that $\sqrt{a^2} = a$ and $\sqrt{-a^2} = ai$.
• Combine like terms: When combining expressions, remember to combine like terms, such as combining the real parts and the imaginary parts separately.
• Check your work: Always check your work to make sure that you have simplified the expression correctly.

Common Mistakes

Here are some common mistakes to avoid when simplifying complex numbers:

• Forgetting to simplify square roots: Make sure to simplify the square roots before combining the expressions.
• Not combining like terms: Make sure to combine like terms, such as combining the real parts and the imaginary parts separately.
• Not checking your work: Always check your work to make sure that you have simplified the expression correctly.

Conclusion

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, which satisfies the equation $i^2 = -1$.

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 a real part and an imaginary part.

Q: How do I simplify a complex number?

A: To simplify a complex number, you need to simplify the square roots and combine the expressions. Remember to use the properties of square roots, combine like terms, and check your work to make sure that you have simplified the expression correctly.

Q: What is the imaginary unit?

A: The imaginary unit, denoted by $i$, is a mathematical constant that satisfies the equation $i^2 = -1$. It is used to represent the imaginary part of a complex number.

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 $c + di$, their 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 $c + di$, their 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 $c + di$, their 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 $c + di$, 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 $a - bi$. It is used to simplify complex fractions and to find the quotient of two complex numbers.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to multiply the numerator and the denominator by the conjugate of the denominator. For example, if you have a complex fraction $\frac{a + bi}{c + di}$, you can simplify it by multiplying the numerator and the denominator by the conjugate of the denominator, which is $c - di$.

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 square roots
• Not combining like terms
• Not checking your work
• Not using the properties of square roots and the distributive property
• Not multiplying the numerator and the denominator by the conjugate of the denominator when dividing complex numbers

Conclusion

In conclusion, complex numbers are an important topic in mathematics, and understanding how to simplify and manipulate them is crucial for success in algebra and beyond. By following the tips and tricks outlined in this article, you can become proficient in working with complex numbers and tackle even the most challenging problems with confidence.