# Square root of 108

The value of root 108 is 10.392. Note that the square root of a number x is denoted by √x, so the square root of 108 can be written as √108. In this section, we will learn how to calculate the square root of 108. But before we do that, here are the few things to remember about the number 108.

108 is a composite even number.

The number 108 is divisible by 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54 and 108.

As 108=9×12, the number 108 is equal to 9 dozen.

Also Read:  Properties of square roots

#### Simplify square root of 108?

What is the square root of 108 in simplest radical form? We will express 108 as a product of perfect squares or as a product of a perfect square and  a non-perfect square.

Note that 108=36×3. Observe that here 36 is a perfect square number whose square root is 6 and the number 3 is a non-perfect square.

Taking square root on both sides of 108 = 36×3, we get that

$\sqrt{108}=\sqrt{36 \times 3}$

$=\sqrt{36} \times \sqrt{3}$  $[\because \sqrt{a \times b}=\sqrt{a} \times \sqrt{b}]$

$=6 \times \sqrt{3}=6\sqrt{3}$

So the simplest radical form of the square root of 108 is 6√3.

#### Is 108 a perfect square number?

We have computed that √108=6√3. So the square root of 108 is not an integer. This makes that 108 is not a perfect square number.

Therefore, 108 is a non-perfect square number. So we deduce that its square root √108 is a quadratic surd.

#### Is Square root of 108 Rational?

As √108=6√3 and the square root of 3 is not a rational number,  so we deduce that the square root of 108 is not a rational number. Note that √108 is an irrational number.

#### What is the value of root 108?

Note that the simplest radical form of square root of 108 is 6√3. Now using the fact that the value of √3 is 1.732, we get that

√108 = 6√3 = 6 × 1.732 = 10.392

So 10.392 is the value of the square root of 108.

#### Square root of 108 by Prime Factorization

The prime factorization method is one of the useful methods to find the square root of a number. Here we will find the square root of 108 by the prime factorization method. At first, we have to factorize the number 108.

As 108 is an even number, it will be divisible by 2. So we have 108=2×54.

Now we will factorize 54 and by the above logic, we get that

54=2×27.

We know that 27=3×9 and 9=3×3. So finally we get that

108=2×2×3×3×3 …(∗)

The above is the prime factorization of 108. Taking square root on both sides of (∗), we get that

$\sqrt{108}=\sqrt{2 \times 2 \times 3 \times 3 \times 3}$

After making pairs of two equal numbers, we have

$\sqrt{108}=\sqrt{2 \times 2} \times \sqrt{3 \times 3} \times \sqrt{3}$

$=2 \times 3 \times \sqrt{3}$ $[\because \sqrt{a \times a}=a]$

$=6\sqrt{3}$

∴  the value of the square root of $108$ is $6\sqrt{3}.$

#### Important Things:

• √108 = 10.392
• 1081/2 is the exponential form of square root of 108.
• √108 is the radical form of square root of 108.
• The square root of 108 is 10.392 corrected up to 3 decimal places.
• Square of 108: 108×108 = 11,664
• 108 is not a perfect square.
• √108 is a quadratic surd.
• The simplified form of √108 is 6√3.