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.