# Simplifying Square Roots Overview

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.

Simplifying Square Root Sample Questions

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.

Question #1:

Write $$\sqrt{175}$$ in the most simplified form possible.

$$5\sqrt{7}$$
$$7\sqrt{5}$$
$$25\sqrt{7}$$
$$35\sqrt{5}$$

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, $$\sqrt{175}=5\sqrt{7}$$.

Question #2:

Which shows $$\sqrt{208}$$ in its most simplified form?

$$2\sqrt{52}$$
$$4\sqrt{52}$$
$$4\sqrt{13}$$
$$13\sqrt{4}$$

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, $$\sqrt{208}=4\sqrt{13}$$.

Question #3:

Simplify $$\sqrt{294}$$.

$$6\sqrt{7}$$
$$7\sqrt{6}$$
$$14\sqrt{21}$$
$$21\sqrt{14}$$

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, $$\sqrt{294}=7\sqrt{6}$$.

Question #4:

Here is an expression: $$\sqrt{648}$$. Which shows the expression in its most simplified form?

$$2\sqrt{18}$$
$$9\sqrt{8}$$
$$12\sqrt{54}$$
$$18\sqrt{2}$$

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, $$\sqrt{648}=18\sqrt{2}$$.

Question #5:

Which shows $$\sqrt{2{,}448}$$ in its most simplified form?

$$4\sqrt{51}$$
$$12\sqrt{17}$$
$$17\sqrt{12}$$
$$48\sqrt{17}$$
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, $$\sqrt{2{,}448}=12\sqrt{17}$$.