Square root of 108

108 can be expressed as 108 = 2² × 3³. The square root of 108 can be written as √108. This is raising 108 to the power ½. In this mini lesson, let us learn about the square root of 108, find out whether the square root of 108 is rational or irrational, and see how to find the square root of  108 by long division method. 

• Square Root of 108: √108 = 10.392
• Square of 108: 108² = 11,664

Finding the square root of a number, say 'n' is finding out what number, say 'a' multiplied by itself equals the number 'n'. a × a = n ⇒ a² = n. Thus a = √n.   √108 =  √(a × a) 

• √108 = √(10.392 × 10.392) or √(-10.392 × -10.392) ⇒ √108 = ±10.392 
• We know that 108 = 2 × 2 × 3 × 3 × 3  In the simplest radical form √108 =  √(2 × 2 × 3 × 3 × 3) =  6√3

The square root of 108 is an irrational number where the numbers after the decimal point go up to infinity. √108 = 10.3923048454. √108 cannot be written in the form of p/q, hence it is an irrational number with never -ending digits. The square root of any number has two values, one is positive and the other is negative. So, √108 = +10.392 and - 10.392.

The square root of 108 or any number can be calculated in many ways. Two of them are the approximation method and the long division method.

Square Root of 108 by Approximation Method

• Take two perfect square numbers which are just smaller than 108 and just greater than 108. It is hence given as, √100 < √108 < √121 ⇒ 10 <  √ 108 < 11
• Using the average method, divide 108 by 10 or 11.
• Let us divide by 11 ⇒ 108 ÷ 11 = 9.81
• Find the average of 9.81 and 11.
• (9.81 + 11) / 2 = 20.81 ÷ 2 = 10.405
• √108 ≈  10.40 

Square Root of 108 by the Long Division Method

The long division method helps us to find a more accurate value of square root of any number.

Let's see how to find the square root of 108 by the long division method.

• Write 108.00 00 00. Take the number in pairs from the right. 08 is chosen as a pair and one stands alone. Now divide 1 with a number such that number × number gives 1 or a number lesser than that.
• Obtain 1 as the quotient and 0 as the remainder. Bring down 08 for division. Double the quotient. It's 2 and have 20  as the new divisor.
• We cannot find a number that divides 08. Hence bring down the next pair of zeros. We have 8 00 as the new dividend. Double the quotient. 200  is the new divisor. 
• Find a number which can be added to 200 and this sum can be multiplied by that number such that the product is 8 00 or lesser than that.
• We find that 3 is such number. 200 + 3 = 203 and 203 × 3 gives 609. Subtract this from 8 00.
• Now our quotient is 10.3 and the remainder is 191. Bring down the next pair of zeros.
• Repeat the process until you get the square root is evaluated to 3 decimal places.
• Thus we evaluate √108 = 10.392

Explore Square roots using illustrations and interactive examples: 

• Square root of 98
• Square root of 54
• Square root of 74
• Square root of 180
• Square root of 120

The square root of any number can be assumed to be between the square root of the two nearest perfect squares of that number. For example, the square root of 108 lies between the square root of 100 and 121.  Therefore, 10 < √108 < 11. Use the average method then, to evaluate the approximate value of √108.

What is the square root of 108?

The square root of 108 is √108 = ±10.392.

What is the square root of 108 simplified?

The square root of 108 is simplified as 6√3.

How to get the square root of 108?

The square root of 108 is got by long division method. 

Is √108 a rational number?

 √108 is an irrational number, because the value of √108 is a non-teminating decimal.

How to find the square root of 108 to the nearest hundredth?

The square root of 108 is evaluated using the division method and rounded of to the nearest hundredth as √108 = 10.39.