Square Root of 106
106 is a composite number as it has more than 2 factors. The other two factors are 2 and 53, which cannot be simplified any further, making √106 an irrational number. In this chapter, we will calculate the square root of 106 by long division method along with solved examples. Let us see what the square root of 106 is.
 Square Root of 106: √106 = 10.295
 Square of 106: 106^{2} = 11236
What Is the Square Root of 106?
Square root is just an inverse operation of square. The number whose square gives 106, is the square root of 106. The square root of 106 is represented as √106.
Is the Square Root of 106 Rational or Irrational?
Square root of 106 cannot be written in the form of p/q, where p and q are integers and q is not equal to 0. The value of √106 is 10.295630140987.. Hence, √106 is not a rational number.
Important Notes:
 106 lies between two perfect square numbers, 100 and 121. Hence, the square root of √106 lies between 10 and 11.
 106 is not a perfect square, hence, √106 is an irrational number.
How to Find the Square Root of 106?
There are different methods to find the square root of any number. Click here to know more about the different methods.
Simplified Radical Form of Square Root of 106
106 is a composite number obtained by the product of two prime numbers, 2 and 53. Hence, the simplified radical form of √106 is √106.
We can find the square root of 106 by the following two methods:
 Prime Factorization Method
 Long Division Method
Square Root of 106 by Prime Factorization
106 can be factorized as a product of 2 and 53, which are prime numbers. Hence, √106 = √(2 × 53). 2 and 53 cannot be factorized any further. Thus, the square root of 106 is written as √106.
Square Root of 106 by Long Division
The value of square root of 106 by long division method consists of the following steps:
 Step 1: First we pair the digits of 106 starting with a digit at one's place. Put a horizontal bar to indicate pairing.
 Step 2: Now we find a number which on multiplication with itself gives a product of less than or equal to 1. As we know 1 × 1 = 1 = 1.
 Step 3: Now, we have to bring down 06 and multiply the quotient by 2. This give us 2. Hence, 2 is the starting digit of the new divisor.
 Step 4: 0 is placed at one's place of new divisor because when 20 is multiplied by 0 we get 0. The obtained answer now is 20 and we bring down 00.
 Step 5: The quotient is now 10 and it is multiplied by 2. This gives 20, which becomes the starting digit of the new divisor.
 Step 6: 2 is placed at one's place of new divisor because on multiplying 202 by 2 we get 404. The answer now obtained is196 and we bring 00 down.
 Step 7: Now the quotient is 102 when multiplied by 2 which gives 204, which will be the starting digit of the new divisor.
 Step 8: 9 is placed at one's place of the divisor because on multiplying 2049 by 9 we get 18441. The answer obtained is 1159 and we bring 00 down.
 Step 9: Now the quotient is 1029 when multiplied by 2 gives 2058, which will be the starting digit of the new divisor.
 Step 10: 5 is placed at one's place of the divisor because on multiplying 20585 by 5, we will obtain 102925. The answer obtained is 12975 and we bring 00 down.
On repeating the above steps we will obtain value of square root of 106 as √106 = 10.295630140987..
Explore square roots using illustrations and interactive examples
Think Tank:
 Can you find any quadratic equation which has a root as √106?
 As (√106)^{2}=106, can we say that √106 is also a square root of 106?
Square Root of 106 Solved Examples

Example 1: What is the length of diagonal for a square having length of each side as 106 units?
Solution
Given, Side of the square = 106 units
Using Pythagoras Theorem, we get
Diagonal of a square = √2a
Diagonal = √(2 × 106) = 14.56 units 
Example 2: What is the difference between lengths of radii of circles having areas 106π and 81π square inches?
Solution
The length of radius for circle having area 106π is: area = πr^{2} = 106π.
Here, r = √106 = 10.29 inches
The length of radius for circle having area 81π is, area = πr^{2} = 81π.
Here, r = √81 = 9 inchesHence, the difference between lengths of radii of circles having areas 106π and 81π square inches is, (10.29  9) = 1.29 inches.
FAQs on Square Root of 106
Is the square root of 106 infinite?
The decimal expansion of √106 is infinite because it results in nonterminating and nonrepeating decimal number.
What is the square root of 106 rounded to its nearest tenth?
The square root of 106 rounded to its nearest tenth is √106 = 10.3.
Why is √106 an irrational number?
A number with decimal expansion as nonterminating and nonrepeating is always an irrational number. So, √106 is an irrational number.
Is the square root of 106 rational or irrational?
The square root of 106 is irrational.
Is square root of 106 a real number?
Yes, the square root of 106 is a real number.