### Learning Objectives

- Define square root
- Find square roots

We know how to square a number:

[latex]5^2=25[/latex] and [latex]\left(-5\right)^2=25[/latex]

Taking a square root is the opposite of squaring so we can make these statements:

- 5 is the nonngeative square root of 25
- -5 is the negative square root of 25

Find the square roots of the following numbers:

- 36
- 81
- -49
- 0

- We want to find a number whose square is 36. [latex]6^2=36[/latex] therefore, the nonnegative square root of 36 is 6 and the negative square root of 36 is -6
- We want to find a number whose square is 81. [latex]9^2=81[/latex] therefore, the nonnegative square root of 81 is 9 and the negative square root of 81 is -9
- We want to find a number whose square is -49. When you square a real number, the result is always positive. Stop and think about that for a second. A negative number times itself is positive, and a positive number times itself is positive. Therefore, -49 does not have square roots, there are no real number solutions to this question.
- We want to find a number whose square is 0. [latex]0^2=0[/latex] therefore, the nonnegative square root of 0 is 0. We do not assign 0 a sign, so it has only one square root, and that is 0.

The notation that we use to express a square root for any real number, a, is as follows:

#### Writing a Square Root

The symbol for the square root is called a **radical symbol.** For a real number, *a* the square root of *a* is written as [latex]\sqrt{a}[/latex]

The number that is written under the radical symbol is called the **radicand**.

By definition, the square root symbol, [latex]\sqrt{\hphantom{5}}[/latex] always means to find the nonnegative root, called the **principal root**.

[latex]\sqrt{-a}[/latex] is not defined, therefore [latex]\sqrt{a}[/latex] is defined for [latex]a>0[/latex]

Let’s do an example similar to the example from above, this time using square root notation. Note that using the square root notation means that you are only finding the principal root – the nonnegative root.

### Example

Simplify the following square roots:

- [latex]\sqrt{16}[/latex]
- [latex]\sqrt{9}[/latex]
- [latex]\sqrt{-9}[/latex]
- [latex]\sqrt{5^2}[/latex]

The last problem in the previous example shows us an important relationship between squares and square roots, and we can summarize it as follows:

#### The square root of a square

For a nonnegative real number, a, [latex]\sqrt{a^2}=a[/latex]

In the video that follows, we simplify more square roots using the fact that [latex]\sqrt{a^2}=a[/latex] means finding the principal square root.

## Summary

The square root of a number is the number which, when multiplied by itself, gives the original number. Principal square roots are always positive and the square root of 0 is 0. You can only take the square root of values that are nonnegative. The square root of a perfect square will be an integer.