Square Roots
Learning Outcomes
- Define and evaluate square roots
- Estimate square roots that are not perfect
- [latex]36[/latex]
- [latex]81[/latex]
- [latex]-49[/latex]
- [latex]0[/latex]
- We want to find a number whose square is [latex]36[/latex]. [latex]6^2=36[/latex] therefore, the nonnegative square root of [latex]36[/latex] is [latex]6[/latex] and the negative square root of [latex]36[/latex] is [latex]-6[/latex]
- We want to find a number whose square is [latex]81[/latex]. [latex]9^2=81[/latex] therefore, the nonnegative square root of [latex]81[/latex] is [latex]9[/latex] and the negative square root of [latex]81[/latex] is [latex]-9[/latex]
- We want to find a number whose square is [latex]-49[/latex]. 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, [latex]-49[/latex] does not have square roots, there are no real number solutions to this question.
- We want to find a number whose square is [latex]0[/latex]. [latex]0^2=0[/latex] therefore, the nonnegative square root of [latex]0[/latex] is [latex]0[/latex]. We do not assign [latex]0[/latex] a sign, so it has only one square root, and that is [latex]0[/latex].
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-display]\sqrt{-a}[/latex] is not defined, therefore [latex]\sqrt{a}[/latex] is defined for [latex]a>0[/latex-display][latex] \begin{array}{r}5\cdot 5=25\\-5\cdot -5=25\end{array}[/latex]
By definition, the square root symbol always means to find the positive root called the principal root. So while [latex]5\cdot5[/latex] and [latex]−5\cdot−5[/latex] both equal [latex]25[/latex], only [latex]5[/latex] is the principal root. You should also know that zero is special because it has only one square root: itself (since [latex]0\cdot0=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]
Answer:
- [latex]\sqrt{16}[/latex]. We are looking for a number whose square is [latex]16[/latex], so [latex]\sqrt{16}=4[/latex]. We only write the nonnegative root because that is how the root symbol is defined.
- [latex]\sqrt{9}[/latex]. We are looking for a number whose square is [latex]9[/latex], so [latex]\sqrt{9}=3[/latex]. We only write the nonnegative root because that is how the root symbol is defined.
- [latex]\sqrt{-9}[/latex]. We are looking for a number whose square is [latex]-9[/latex]. There are no real numbers whose square is [latex]-9[/latex], so this radical is not a real number.
- [latex]\sqrt{5^2}[/latex]. We are looking for a number whose square is [latex]5^2[/latex]. We already have the number whose square is [latex]5^2[/latex], it's [latex]5[/latex]!
The square root of a square
For a nonnegative real number, a, [latex]\sqrt{a^2}=a[/latex]Radical | Name | Simplified Form |
---|---|---|
[latex] \sqrt{36}[/latex] | “Square root of thirty-six” “Radical thirty-six” | [latex] \sqrt{36}=\sqrt{6^2}=6[/latex] |
[latex] \sqrt{100}[/latex] | “Square root of one hundred” “Radical one hundred” | [latex] \sqrt{100}=\sqrt{10^2}=10[/latex] |
[latex] \sqrt{225}[/latex] | “Square root of two hundred twenty-five” “Radical two hundred twenty-five” | [latex] \sqrt{225}=\sqrt{15^2}=15[/latex] |
Try It
[ohm_question]228[/ohm_question]Finding Square Roots Using Factoring
What if you are working with a number whose square you do not know right away? We can use factoring and the product rule for square roots to find square roots such as [latex]\sqrt{144}[/latex], or [latex]\sqrt{225}[/latex].The Product Rule for Square Roots
Given that a and b are nonnegative real numbers, [latex]\sqrt{a\cdot{b}}=\sqrt{a}\cdot\sqrt{b}[/latex]- square root of a square,
- the product rule for square roots
- factoring
Example
Simplify [latex] \sqrt{144}[/latex]Answer: Determine the prime factors of [latex]144[/latex].
[latex]\begin{array}{c}\sqrt{144}\\\\\sqrt{2\cdot 72}\\\\\sqrt{2\cdot 2\cdot 36}\\\\\sqrt{2\cdot 2\cdot 2\cdot 18}\\\\\sqrt{2\cdot 2\cdot 2\cdot 2\cdot 9}\\\\\sqrt{2\cdot 2\cdot 2\cdot 2\cdot 3\cdot 3}\end{array}[/latex]
Because we are finding a square root, we regroup these factors into squares.[latex]\sqrt{2^2\cdot 2^2\cdot3^2}[/latex]
Now we can use the product rule for square roots and the square root of a square idea to finish finding the square root.[latex]\begin{array}{c}\sqrt{2^2\cdot 2^2\cdot3^2}\\\\=\sqrt{2^2}\cdot\sqrt{2^2}\cdot\sqrt{3^2}\\\\=2\cdot3\cdot2\\\\=12\end{array}[/latex]
Answer
[latex-display] \sqrt{144}=12[/latex-display]Example
Simplify [latex]\sqrt{225}[/latex]Answer: First, factor [latex]225[/latex]:
[latex]\begin{array}{c}\sqrt{225}\\\\=\sqrt{5\cdot45}\\\\=\sqrt{5\cdot5\cdot9}\\\\=\sqrt{5\cdot5\cdot3\cdot3}\end{array}[/latex]
Because we are finding a square root, we regroup these factors into squares. Finish simplifying with the product rule for roots, and the square of a square idea.[latex]\begin{array}{c}\sqrt{5^2\cdot3^2}\\\\=\sqrt{5^2}\cdot\sqrt{3^2}\\\\=5\cdot3=15\end{array}[/latex]
Answer
[latex-display]\sqrt{225}=15[/latex-display][latex]\sqrt{a+b}\ne\sqrt{a}+\sqrt{b}[/latex]
Prove this to yourself with some real numbers: let [latex]a = 64[/latex] and [latex]b = 36[/latex], then use the order of operations to simplify each expression.
[latex]\begin{array}{c}\sqrt{64+36}=\sqrt{100}=10\\\\\sqrt{64}+\sqrt{36}=8+6=14\\\\10\ne14\end{array}[/latex]
Example
Find the principal root of each expression.- [latex]\sqrt{100}[/latex]
- [latex]\sqrt{16}[/latex]
- [latex]\sqrt{25+144}[/latex]
- [latex]\sqrt{49}-\sqrt{81}[/latex]
- [latex] -\sqrt{81}[/latex]
- [latex]\sqrt{-9}[/latex]
Answer:
- [latex]\sqrt{100}=10[/latex] because [latex]{10}^{2}=100[/latex]
- [latex]\sqrt{16}=4[/latex] because [latex]{4}^{2}=16[/latex]
- Recall that square roots act as grouping symbols in the order of operations, so addition and subtraction must be performed first when they occur under a radical. [latex]\sqrt{25+144}=\sqrt{169}=13[/latex] because [latex]{13}^{2}=169[/latex]
- This problem is similar to the last one, but this time subtraction should occur after evaluating the root. Stop and think about why these two problems are different. [latex]\sqrt{49}-\sqrt{81}=7 - 9=-2[/latex] because [latex]{7}^{2}=49[/latex] and [latex]{9}^{2}=81[/latex]
-
The negative in front means to take the opposite of the value after you simplify the radical. [latex] -\sqrt{81}\\-\sqrt{9\cdot 9}[/latex]. The square root of [latex]81[/latex] is [latex]9[/latex]. Then, take the opposite of [latex]9[/latex] to get [latex]−(9)[/latex]
- We are looking for a number that when it is squared returns [latex]-9[/latex]. We can try [latex](-3)^2[/latex], but that will give a positive result, and [latex]3^2[/latex] will also give a positive result. This leads to an important fact - you cannot find the square root of a negative number.
Think About It
Does [latex]\sqrt{25}=\pm 5[/latex]? Write your ideas and a sentence to defend them in the box below before you look at the answer. [practice-area rows="1"][/practice-area]Answer: No. Although both [latex]{5}^{2}[/latex] and [latex]{\left(-5\right)}^{2}[/latex] are [latex]25[/latex], the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is [latex]\sqrt{25}=5[/latex].
Example
Simplify. [latex] \sqrt{63}[/latex]Answer: Factor [latex]63[/latex]
[latex] \sqrt{7\cdot 3\cdot3}[/latex]
Regroup factors into squares
[latex] \sqrt{7\cdot3^2}[/latex]
Finish simplifying with the product rule for roots, and the square of a square idea.
[latex]\sqrt{7\cdot3^2}\\\\=\sqrt{7}\cdot\sqrt{3^2}\\\\=\sqrt{7}\cdot3[/latex]
Since 7 is prime and we can't write it as a square, it will have to stay under the radical sign. As a matter of convention, we write the constant, [latex]3[/latex], in front of the radical. This helps the reader know that the [latex]3[/latex] is not under the radical anymore.
[latex] 3\cdot \sqrt{7}[/latex]
Answer
[latex-display] \sqrt{63}=3\sqrt{7}[/latex-display]- [latex]0^2=0[/latex]
- [latex]2^2=4[/latex]
- [latex]3^2=9[/latex]
- [latex]4^2=16[/latex]
- [latex]5^2=25[/latex]
- [latex]6^2=36[/latex]
- [latex]7^2=49[/latex]
- [latex]8^2=64[/latex]
- [latex]9^2=81[/latex]
- [latex]10^2=100[/latex]
Example
Simplify. [latex] \sqrt{2,000}[/latex]Answer: Factor [latex]2,000[/latex] to find perfect squares. [latex-display] \begin{array}{r}\sqrt{100\cdot 20}\\\\=\sqrt{100\cdot 4\cdot 5}\end{array}[/latex-display] [latex-display]100=10^2,4=2^2[/latex-display]
[latex]\begin{array}\sqrt{100\cdot 4\cdot 5}\\\\=\sqrt{10^2\cdot 4^2\cdot 5}\\\\=\sqrt{10^2}\cdot \sqrt{4^2}\cdot \sqrt{5}\\\\=10\cdot 4\cdot \sqrt{5}\end{array}[/latex]
Multiply.[latex] 20\cdot \sqrt{5}[/latex]
Answer
[latex-display] \sqrt{2,000}=20\sqrt{5}[/latex-display]Estimate Roots
An approach to handling roots that are not perfect squares is to approximate them by comparing the values to perfect squares. Suppose you wanted to know the square root of [latex]17[/latex]. Let us look at how you might approximate it.Example
Estimate [latex] \sqrt{17}[/latex]Answer: Think of two perfect squares that surround [latex]17[/latex]. [latex]17[/latex] is in between the perfect squares [latex]16[/latex] and [latex]25[/latex]. So, [latex] \sqrt{17}[/latex] must be in between [latex] \sqrt{16}[/latex] and [latex] \sqrt{25}[/latex]. Determine whether [latex] \sqrt{17}[/latex] is closer to [latex]4[/latex] or to [latex]5[/latex] and make another estimate.
[latex] \sqrt{16}=4[/latex] and [latex] \sqrt{25}=5[/latex]
Since [latex]17[/latex] is closer to [latex]16[/latex] than [latex]25[/latex], [latex] \sqrt{17}[/latex] is probably about [latex]4.1[/latex] or [latex]4.2[/latex]. Use trial and error to get a better estimate of [latex] \sqrt{17}[/latex]. Try squaring incrementally greater numbers, beginning with [latex]4.1[/latex], to find a good approximation for [latex] \sqrt{17}[/latex].[latex]\left(4.1\right)^{2}[/latex]
[latex]\left(4.1\right)^{2}[/latex] gives a closer estimate than [latex](4.2)^{2}[/latex].[latex]4.1\cdot4.1=16.81\\4.2\cdot4.2=17.64[/latex]
Continue to use trial and error to get an even better estimate.[latex]4.12\cdot4.12=16.9744\\4.13\cdot4.13=17.0569[/latex]
[latex-display] \sqrt{17}\approx 4.12[/latex-display]Try It
[ohm_question]146633[/ohm_question]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 [latex]0[/latex] is [latex]0[/latex]. You can only take the square root of values that are nonnegative. The square root of a perfect square will be an integer. Other square roots can be simplified by identifying factors that are perfect squares and taking their square root. Square roots that are not perfect can also be estimated by finding the perfect square above and below your number. Also, using a calculator is useful for finding a more precise approximation of a square root.Contribute!
Licenses & Attributions
CC licensed content, Original
- Image: Shortcut this way.. Provided by: Lumen Learning License: CC BY: Attribution.
- Simplify Square Roots (Perfect Square Radicands). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify Square Roots (Not Perfect Square Radicands). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education Located at: https://www.nroc.org/. License: CC BY: Attribution.