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.

**Answer:**

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.

**Answer:**

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 $\sqrt 2$.

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