Square root of 11
Square root of 11 can be determined using long division method. Square root of 11 is an irrational number. In this lesson we will find how to calculate square root of 11 and why the square root of 11 is an irrational number. Let us see what the square root of 11 is.
 Square Root of 11: √11 = 3.3166
 Square of 11: 11^{2} = 121
1.  What Is the Square Root of 11? 
2.  Is Square Root of 11 Rational or Irrational? 
3.  How to Find the Square Root of 11? 
4.  FAQs on Square Root of 11 
What Is the Square Root of 11?
The square root is an inverse mathematical operation of a square. The square root of a number is the number that gets multiplied to itself to give the product. Let's take a look at the below example: This is a number which when multiplied to itself gives us the result as 11. The square root of 11 is represented as √11. The square root of 11 in the exponent form is expressed as 11^{½ }. The square root of 11 rounded to 5 decimal places is + 3.31662 and  3.31662.
Is the Square Root of 11 Rational or Irrational?
A rational number is a number that is of the form p/q where:
 p and q are integers
 q ≠ 0
A number that cannot be expressed as a ratio of two integers is an irrational number. Nonterminating decimals having repeated numbers after the decimal point are rational numbers. Now let us look at the square root of 11
 √11 = 3.31662479036
Do you think the decimal part stops after 3.31662479036? Clearly No, this is nonterminating and the decimal part has no repeating pattern. So it is not a rational number. Thus, √11 is an irrational number.
How to Find the Square Root of 11?
Square Root of 11 can be calculated using various methods:
 By simplifying the radical of the numbers that are perfect squares.
 By long division method for perfect and nonperfect squares
11 is a prime number and hence, it is not a perfect square.
Therefore, the square root of 11 can only be found by the long division method.
Simplified Radical Form of Square Root of 11
To simplify the square root of 11, let us first express 11 as a product of its prime factors. The prime factorization of 11 can be found as follows: Thus the prime factorization of 11 is: 1 × 11 To find the square root of any number, we take one number from each pair of the same numbers from its prime factorization and we multiply them. But the prime factorization of 11 is 11 × 11 which has no pairs of the same numbers. Therefore, √11 is in the lowest form and cannot be simplified further. Can you try and express the square root of 5 and the square root of 10 in a similar way?
Square Root of 11 By Long Division
The value of the square root of 11 by long division method consists of the following steps:
 Step 1: Starting from the right, we will pair up the digits by putting a bar above them.
 Step 2: Find a number such that when you multiply it with itself, the product is less than or equal to 11. So, the number is 3. Putting the divisor as 3, we get the quotient as 3 and the remainder 2
 Step 3: Double the divisor and enter it with a blank on its right. Guess the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend. Divide and write the remainder. Repeat this process to get the decimal places you want.
Hence, √11 = 3.3166
Explore Square roots using illustrations and interactive examples
Think Tank:
 Can you think of any quadratic equation which has a root as √11?
 Is √11 a real number?
 Since (√11)^{2 }= 11, can we say that √11 is also a square root of 11?
Important Notes:
 The real roots of √11 are ± 3.316.
 Irrational numbers cannot be expressed as a ratio of two integers. Example: π, √2.
Square Root of 11 Solved Examples

Example 1: Jim had a doubt. He knew that 3.316 is the square root of 11. He wanted to know if 3.316 is also a square root of 11? Can you clarify his doubt?
Solution
Let us take an example of a perfect square number and extend the same logic to clarify her doubt. We know that 4 is a square root of 16 because when 4 is multiplied to itself it gives 16 But what about 4?
Let us multiply and check (4) × (4) = 16, () × () = +
Therefore, 4 is also a ssquare root of 16. Going by the same logic, 3.316 is also a square root of 11. 
Example 2: John is baking a square cake for his friend's birthday party. He decides to give each guest a square piece of cake of side 1inch. There are 121 guests attending the party. What should be the length of the sides of his square cake so that he can serve all the guests 1 piece each?
Solution
Since there are 121 guests, the number of cake pieces should be 121. Since each piece of cake has to be 1 inch by 1 inch, the area of the square cake should be 121 sq. inches.
Therefore, length of the sides of the cake = 121 = √11
Length of the sides of the square cake = 11 inches
FAQs On Square Root of 11
What is the square root of 11?
The square root of 11 is 3.3166. √11 = 3.3166.
What is the square of 11?
The square of 11 is 121
11^{2} = 121
How do you work out the square root of 11?
We can find the square root of 11 using various methods.
 Repeated Subtraction
 Prime Factorization
 Estimation and Approximation
 Long Division
If you want to learn more about each of these methods, click here.
Is square root of 11 a real number?
Yes, the square root of 11 is a real number.
What is the square root of 11 in the simplest radical form?
√11 is the simplest radical form.
Is the square root of 11 a rational number?
As the value of √11 is 3.3166. Hence, √11 is an irrational number.