Which Expression Is Equivalent To $\sqrt{-80}$?A. $-4 \sqrt{5}$ B. $-4i \sqrt{5}$ C. $4i \sqrt{5}$ D. $4 \sqrt{5}$

Simplifying Radical Expressions: A Guide to Understanding Imaginary Numbers

Radical expressions are a fundamental concept in mathematics, and they can be simplified using various techniques. However, when dealing with negative numbers under the square root, things can get a bit more complicated. In this article, we will explore the concept of imaginary numbers and how to simplify radical expressions involving negative numbers.

Imaginary numbers are a type of complex number that, when squared, give a negative result. They are denoted by the letter "i" and are used to extend the real number system to the complex number system. In other words, imaginary numbers are the square roots of negative numbers.

The Concept of i

The imaginary unit "i" is defined as the square root of -1. This means that i^2 = -1. Using this definition, we can simplify radical expressions involving negative numbers.

To simplify radical expressions involving negative numbers, we can use the concept of imaginary numbers. When we see a negative number under the square root, we can rewrite it as the product of a positive number and the imaginary unit "i".

Example 1: Simplifying $\sqrt{-80}$

Let's consider the expression $\sqrt{-80}$. We can rewrite this expression as $\sqrt{-1} \cdot \sqrt{80}$.

Step 1: Simplify the square root of 80

The square root of 80 can be simplified as $\sqrt{80} = \sqrt{16 \cdot 5} = 4 \sqrt{5}$.

Step 2: Simplify the square root of -1

The square root of -1 is equal to the imaginary unit "i".

Step 3: Combine the results

Using the results from steps 1 and 2, we can simplify the expression $\sqrt{-80}$ as $i \cdot 4 \sqrt{5} = 4i \sqrt{5}$.

In conclusion, when dealing with radical expressions involving negative numbers, we can use the concept of imaginary numbers to simplify them. By rewriting the negative number as the product of a positive number and the imaginary unit "i", we can simplify the expression and arrive at the final result.

The correct answer is B. $-4i \sqrt{5}$ is incorrect, the correct answer is C. $4i \sqrt{5}$.

The answer C is correct because we simplified the expression $\sqrt{-80}$ as $i \cdot 4 \sqrt{5} = 4i \sqrt{5}$. This is the correct result, and it matches the answer C.

The answer A is incorrect because we simplified the expression $\sqrt{-80}$ as $i \cdot 4 \sqrt{5} = 4i \sqrt{5}$, not $-4 \sqrt{5}$.

The answer B is incorrect because we simplified the expression $\sqrt{-80}$ as $i \cdot 4 \sqrt{5} = 4i \sqrt{5}$, not $-4i \sqrt{5}$.

The answer D is incorrect because we simplified the expression $\sqrt{-80}$ as $i \cdot 4 \sqrt{5} = 4i \sqrt{5}$, not $4 \sqrt{5}$.

The final answer is C. $4i \sqrt{5}$.
Simplifying Radical Expressions: A Guide to Understanding Imaginary Numbers

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

A: A real number is a number that can be expressed on the number line, such as 3, 4, or -2. An imaginary number, on the other hand, is a number that cannot be expressed on the number line, such as the square root of -1.

Q: How do you simplify a radical expression involving a negative number?

A: To simplify a radical expression involving a negative number, you can rewrite the negative number as the product of a positive number and the imaginary unit "i". For example, $\sqrt{-80}$ can be rewritten as $\sqrt{-1} \cdot \sqrt{80}$.

Q: What is the value of i?

A: The value of i is the square root of -1. This means that i^2 = -1.

Q: How do you simplify a radical expression involving a negative number using i?

A: To simplify a radical expression involving a negative number using i, you can follow these steps:

1. Rewrite the negative number as the product of a positive number and i.
2. Simplify the square root of the positive number.
3. Multiply the result by i.

Q: Can you give an example of simplifying a radical expression involving a negative number using i?

A: Yes, let's consider the expression $\sqrt{-80}$. We can rewrite this expression as $\sqrt{-1} \cdot \sqrt{80}$.

Step 1: Simplify the square root of 80

The square root of 80 can be simplified as $\sqrt{80} = \sqrt{16 \cdot 5} = 4 \sqrt{5}$.

Step 2: Simplify the square root of -1

The square root of -1 is equal to the imaginary unit "i".

Step 3: Combine the results

Using the results from steps 1 and 2, we can simplify the expression $\sqrt{-80}$ as $i \cdot 4 \sqrt{5} = 4i \sqrt{5}$.

Q: What is the final answer for the expression $\sqrt{-80}$?

A: The final answer for the expression $\sqrt{-80}$ is $4i \sqrt{5}$.

Q: Can you give another example of simplifying a radical expression involving a negative number using i?

A: Yes, let's consider the expression $\sqrt{-36}$. We can rewrite this expression as $\sqrt{-1} \cdot \sqrt{36}$.

Step 1: Simplify the square root of 36

The square root of 36 can be simplified as $\sqrt{36} = \sqrt{6^2} = 6$.

Step 2: Simplify the square root of -1

The square root of -1 is equal to the imaginary unit "i".

Step 3: Combine the results

Using the results from steps 1 and 2, we can simplify the expression $\sqrt{-36}$ as $i \cdot 6 = 6i$.

Q: What is the final answer for the expression $\sqrt{-36}$?

A: The final answer for the expression $\sqrt{-36}$ is $6i$.

In conclusion, simplifying radical expressions involving negative numbers can be done using the concept of imaginary numbers. By rewriting the negative number as the product of a positive number and the imaginary unit "i", we can simplify the expression and arrive at the final result.