# How to Simplify Square Roots

The simplification of square roots in their simplest radical form is one of the main topics in Maths. In this post, we will learn how to simplify a square root in lowest radical form.

## What is a square root

Definition of square roots: A number x is called a square root of another number b, if x and b are related by

$x \times x =b$

That is, $x^2=b$.

The square root is represented by the symbol sqrt, and mathematically we can write it as $\sqrt{b} =x$.

## Steps to simplify square roots

The following steps we need to consider in order to simplify the square root of a number $b$ in simplified form:

Step 1: First, we will express $b$ as a product of distinct prime numbers. This is called the prime factorization of $b$.

Step 2: Make pairs of equal prime numbers and apply the rule $\sqrt {a \times a} =a$.

Step 3: After the process explained in Step 2, if there is any prime number $p$ left without a pair, then $\sqrt p$ will be kept as it is.

Step 4: Multiply the numbers obtained in steps 2 and 3, the resulting number will be the square root of the given number $b$.

## Examples of Simplifying Square Roots

Let us now understand the procedure to simplify a square root with an example. In the examples below, we will find the square roots of 4 and 8.

Example 1: Find the square root of 4.

We know that 4 is divisible by 2 and we have 4=2×2. So the prime factorization of 4 is given by

4=2×2

Taking square root on both sides and applying the rule $\sqrt {a \times a} =a$, we get that

$\sqrt 4 = \sqrt{2 \times 2}= 2$.

So 2 is the square root of 4.

Example 2: Find the square root of 8.

As 8 =2×4 and 4=2×2, the prime factorization of 8 will be equal to 8=2×2×2.

Now, taking square roots on both sides, we get that

$\sqrt 8 = \sqrt {2 \times 2 \times 2}$

= $\sqrt{2 \times 2} \times \sqrt{2}$

= $2 \times \sqrt 2$

= $2\sqrt 2$.

So the simplest radical form of square root of 8 is 2 $\sqrt 2$.