# Negative Square Root: Definition & Overview

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• 0:00 What Is a Square Root?
• 1:00 What Is the Negative…
• 2:46 The Imaginary Unit
• 4:17 Operations With…
• 5:57 Powers of the Imaginary Unit
• 7:26 Lesson Summary

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Lesson Transcript
Instructor: David Liano
After completing this lesson, you will know how to write the square root of a negative number. You will also be able to complete mathematical operations that involve square roots of negative numbers.

## What Is A Square Root?

Before we learn what a negative square root is, let's first define what a square root is. The number a is the square root of b in the expression a^2 = b. This means that if you multiply a by itself, or a by a, you will get b. Let's plug real numbers into that equation, where a is 4:

4^2=16

This means that 4 times 4 is 16, and therefore 4 is the square root of 16.

A positive number has two square roots: one is positive and one is negative. If we have a positive number b, then its square roots are written as shown in Figure 1. The negative square root of b has the negative sign.

Let's again look at an actual number. The two square roots of 16 are 4 and -4 because 4^2 = 16 and (-4)^2 = 16 as seen in the following Figure 2.

## What Is The Negative Square Root?

As shown earlier, a negative square root is one of two square roots of a positive number. For the number 25, its negative square root is -5 because (-5)^2 = 25.

We can solve certain equations by finding the square root of a number. Let's consider the equation of x^2 = 121. We want to solve for x, so we need to take the square root of both sides of the equation as shown in Figure 3.

We use the symbol ± because we need to consider both square roots of 121. This symbol is read as 'plus or minus the square root of 121.' The solution to the problem is +11 or -11. We can check this by plugging the answers into the original equation:

11^2 = 121

(-11)^2 = 121

What about the square root of a negative number? For instance, what is the square root of -9? We can try 3, but 3 x 3 = 9. We can try -3, but (-3) x (-3) = 9. This dilemma is due to the fact that the square root of any real number x cannot be negative. Therefore, the square root of a negative number does not exist, at least not within the system of real numbers.

We should recall that real numbers include all the rational numbers (e.g., the whole numbers 0 and 7, the integer -5, and the fraction 2/3) as well as the irrational numbers (like pi and square root of 3).

However, mathematicians overcame this problem of square roots of negative numbers by creating the imaginary unit.

## The Imaginary Unit

The imaginary unit i is defined as the square root of -1.

A primary reason for creating the imaginary unit was for solving quadratic equations that have no real number solutions. Let's consider a simple quadratic equation such as the following:

x^2 + 4 = 0

If we solve for x, we will get x = ± square root of -4. What then are possible values of x?

2 x 2 does not equal -4, and (-2) x (-2) does not equal -4.

This tells us that the graph of this quadratic equation does not have any solutions that are real numbers. In other words, it does not cross the x axis.

However, we can give it imaginary solutions. We can use the product property of square roots and re-write the square root of -4 as shown in Figure 4. We isolated the imaginary number, which gave us a positive number 4 under the other square root symbol. We replace the square root of -1 with i and finish simplifying as normal.

There, we have found the solutions for the quadratic equation x^2 + 4 = 0, albeit they are imaginary solutions.

## Operations With Imaginary Numbers

Now let's look at operations with imaginary numbers and start off with a simple example:

• Simplify the square root of -18 plus the square root of -50.

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