# Square Roots – Examples and Practice Problems

The square root can be considered the opposite operation of squaring a number. With the square root, we are looking for a number that when squared produces the number below the square root. In this article, we will look at a summary of the square root.

In addition, we will see examples with answers to understand how to solve these types of problems.

##### ALGEBRA

Relevant for

Exploring several examples of square roots.

See examples

##### ALGEBRA

Relevant for

Exploring several examples of square roots.

See examples

## Summary of the square roots

Square roots are the opposite of squaring a number or multiplying it by itself. For example, 4 squared equals 16 ($latex {{4}^2}=16$). This means that the square root of 16 equals 4. Using mathematical symbols, we have:

$latex \sqrt{16}=4$

The symbol “√” tells us that we have to take the square root of a number. It is important to remember that all numbers actually have two square roots. For example, four times four equals sixteen, but four negative times four negative also equals sixteen. Therefore, we have

$latex \sqrt{16}= \pm 4$

In some cases, we can ignore the negative square roots of numbers, but sometimes it is important to remember that every number has two square roots.

One of the challenges with square roots can be simplifying large square roots. To do this, we have to follow a few simple rules. We can factor square roots in the same way that we factor numbers. For example, if we have the square root of six, we can write the following:

$latex \sqrt{6}=\sqrt{2} \sqrt{3}$

## Square roots – Examples with answers

These square root examples can be used to master square root problem-solving. Each example has its respective solution, but it is recommended that you try to solve the exercises yourself before looking at the answer. In the following examples, we only take into account the positive square root of the number.

### EXAMPLE 1

Find the following: $latex \sqrt{25}$.

We have to find a number that, when multiplied by itself, produces 25. The answer is 5 since if we multiply 5 by itself, we obtain:

$latex 5\times 5=25$

### EXAMPLE 2

Find the square root of 121: $latex \sqrt{121}$.

We have to find a number that, when multiplied by itself, results in 121. This number is equal to 11 since when we square 11, we obtain:

$latex {{11}^2}=121$

### EXAMPLE 3

Find the following: $latex \sqrt{32}$.

In this case, there is no integer that can be multiplied by itself to obtain 32. However, we can factor this expression and write as follows:

$latex \sqrt{32}=\sqrt{16}\sqrt{2}$

Now, we can find the square root of 16. We know that multiplying by 4 by itself we get 16, so we have:

$latex \sqrt{16}\sqrt{2}=4\sqrt{2}$

### EXAMPLE 4

Simplify the following: $latex \sqrt{50}$.

In this case, there is also no integer which, when multiplied by itself, results in 50. Therefore, we rewrite this square root as follows:

$latex \sqrt{50}=\sqrt{25}\sqrt{2}$

Similar to the previous problem, we can find an integer that results in 25 when squared. This number is 5, so we have:

$latex \sqrt{25}\sqrt{2}=5\sqrt{2}$

### EXAMPLE 5

Simplify the following: $latex \sqrt{132}$.

132 is a big number and it is a bit difficult to know what we can do. However, we can see that it is divisible by 2, so we can write:

$latex \sqrt{132}=\sqrt{66}\sqrt{2}$

We also know that 66 is divisible by 2, so we write:

$latex \sqrt{66}\sqrt{2}=\sqrt{33}\sqrt{2}\sqrt{2}$

If we multiply the square root of a number by itself, we get the original number. Thus, we have:

$latex \sqrt{33}\sqrt{2}\sqrt{2}=2\sqrt{33}$

## Square roots – Practice problems

Practice what you have learned and test your knowledge with the following square root problems. Choose an answer and click “Check” to verify that you selected the correct answer. The solved examples above can serve as a guide if you have difficulty with these problems.

#### Simplify the expression $latex \sqrt{180}$.

Interested in learning more about algebraic topics? Take a look at these pages: ### Jefferson Huera Guzman

Jefferson is the lead author and administrator of Neurochispas.com. The interactive Mathematics and Physics content that I have created has helped many students.  