The square root of a number is a number that when multiplied by itself, results in the original number. For example, \(5\times5=25\), therefore, the square root of \(25\) is \(5\). The radical symbol, \(\sqrt{ }\), is used to represent square roots. The two statements, “What is the square root of \(36\)?” and “What is \(\sqrt{36}\)?” are read the exact same way.
To simplify a square root, we simply find the value that when multiplied by itself gives us the original number. When the square root is a whole number, this is called, perfect square.
Here are some examples of perfect squares:
\(\sqrt{4}=2\)
\(\sqrt{9}=3\)
\(\sqrt{16}=4\)
\(\sqrt{25}=5\)
When a square root does not simplify to a whole number, we simplify what we can and whatever is remaining stays under the radical symbol. For example, to simplify \(\sqrt{28}\), we will look at the factors of \(\sqrt{28}\) to see if there are any perfect squares. \(28=2\times2\times7\) or \(4\times7\). Since \(4\) is a perfect square and \(\sqrt{4}=2\), we can pull that out and we are left with \(7\) under the radical, so \(\sqrt{28}=2\sqrt{7}\).
Example
Simplify \(\sqrt{180}\).
\(180=2\times2\times3\times3\times5\)
Therefore, \(\sqrt{180}=2\times3\sqrt{5}=6\sqrt{5}\).
Simplifying Square Root Sample Questions
Here are a few sample questions going over simplifying square roots.
Write \(\sqrt{175}\) in the most simplified form possible.
To simplify the expression \(\sqrt{175}\), we will start by finding the factors of \(175\), which is \(5\times5\times7\). Since \(5\times5\) is a perfect square, we can take one \(5\) out and we are left with \(7\) under the radical; therefore, \(\sqrt{175}=5\sqrt{7}\).
Which shows \(\sqrt{208}\) in its most simplified form?
The simplified form of \(\sqrt{208}\) can be found by first finding the factors of \(208\), which is \(2\times2\times2\times2\times13\). There are two perfect squares, \(2\times2\), that we can pull out, which leaves us with \(13\) under the radical. Therefore, \(\sqrt{208}=4\sqrt{13}\).
Simplify \(\sqrt{294}\).
We will simplify \(\sqrt{294}\) by first finding the factors of \(294\), which is \(2\times3\times7\times7\). There is one perfect square, \(7\times7\), so we pull one out and we are left with \(2\times3\), or \(6\), under the radical. Therefore, \(\sqrt{294}=7\sqrt{6}\).
Here is an expression: \(\sqrt{648}\). Which shows the expression in its most simplified form?
To find the most simplified form of the expression \(\sqrt{648}\), we will start by finding the factors of \(648\), which is \(2\times2\times2\times3\times3\times3\times3\). There are three sets of perfect squares, \(4,9,\) and \(9\), which we will pull out, and we are going to be left with \(2\) under the radical. Therefore, \(\sqrt{648}=18\sqrt{2}\).
Which shows \(\sqrt{2{,}448}\) in its most simplified form?
The most simplified form of the expression \(\sqrt{2{,}448}\) can be found by first finding the factors of \(2{,}448\), which is \(2\times2\times2\times2\times3\times3\times17\). There are three sets of perfect squares, \(4,4,\) and \(9\), which we will pull out from under the radical, and will be left with \(17\). Therefore, \(\sqrt{2{,}448}=12\sqrt{17}\).