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Last updated on May 26th, 2025

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Square root of 11

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The square root of 11 is the inverse operation of squaring a value “y” such that when “y” is multiplied by itself → y ⤫ y, the result is 11. It contains both positive and a negative root, where the positive root is called the principal square root.

Square root of 11 for US Students
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What Is the Square Root of 11?

The square root of 11 is ±3.31662479036. The positive value,3.31662479036 is the solution of the equation x2 = 11. As defined, the square root is just the inverse of squaring a number, so, squaring 3.31662479036 will result in 11.  The square root of 11 is expressed as √11 in radical form, where the ‘√’  sign is called “radical”  sign. In exponential form, it is written as (11)1/2  
 

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Finding the Square Root of 11

We can find the square root of 11 through various methods. They are:

 

  • Prime factorization method

 

  • Long division method

 

  • Approximation/Estimation method
     
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Square Root of 11 By Prime Factorization Method

The prime factorization of 11 involves breaking down a number into its factors. Divide 11 by prime numbers, and continue to divide the quotients until they can’t be separated anymore. After factorizing 11, make pairs out of the factors to get the square root. If there exists numbers which cannot be made pairs further, we place those numbers with a “radical” sign along with the obtained pairs

 

So, Prime factorization of 11 =11 × 1    

 


for 11, no pairs of factors are obtained, but a single 11 is obtained.

 


So, it can be expressed as  √11 = √(11 × 1) = √11

 


√11 is the simplest radical form of √11

 

 

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Square Root of 11 by Long Division Method

This is a method used for obtaining the square root for non-perfect squares, mainly. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder too sometimes.

 

Follow the steps to calculate the square root of 11:

 


Step 1 : Write the number 11, and draw a bar above the pair of digits from right to left.


               
Step 2 : Now, find the greatest number whose square is less than or equal to 11. Here, it is 3, Because 32=9 < 11

 

Step 3 : Now divide 11 by 3 (the number we got from Step 2) such that we get 3 as quotient, and we get a remainder. Double the divisor 3, we get 6 and then the largest possible number A1=3 is chosen such that when 3 is written beside the new divisor, 6, a 2-digit number is formed →63 and multiplying 3 with 63 gives 189 which is less than 200.

 

Repeat the process until you reach remainder 0


We are left with the remainder, 16444 (refer to the picture), after some iterations and keeping the division till here, at this point 


             
Step 4 : The quotient obtained is the square root. In this case, it is 3.316…

 

 

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Square Root of 11 by Approximation Method

Approximation or estimation of square root is not the exact square root, but it is an estimate.Here, through this method, an approximate value of square root is found by guessing.

 

Follow the steps below:


Step 1 : Identify the square roots of the perfect squares above and below 11.


Below : 9→ square root of 9 = 3     ……..(i)


Above : 16 →square root of 16= 4     ……..(ii)


Step 2 : Divide 11 with one of 3 or 4


 If we choose 3, and divide 11 by 3, we get 3.666   …….(iii)


             
Step 3: Find the average of 3 (from (i)) and 3.666 (from (iii))


(3+3.666)/2 = 3.33

            
Hence, 3.333 is the approximate square root of 11


 

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Common Mistakes and How to Avoid Them in the Square Root of 11

When we find the square root of 11, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Incorrectly applying the square root property
 

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Square root do not distribute over additions. For example, √11 ≠  √5 +  √6
 

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Square Root of 11 Examples

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Problem 1

Simplify 5√11?

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5√11

 

= 5⤬√11

 

= 5⤬3.316

 

= 16.58


Answer : 16.58
 

Explanation

√11= 3.316, so multiplying the square root value with 5

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Problem 2

What is √11 + √11+ √11 ?

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√11+ √11 + √11

 

= 3.316+3.316+3.316

 

=9.948


Answer: 9.948

Explanation

adding the square root value of 11 thrice.
 

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Problem 3

Find the value of 1/√11?

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1/√11

 

= 1/ 3.316

 

=0.30156


Answer: 0.30156

Explanation

we divide 1 by the value of √11
 

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Problem 4

If y=√11, find y²

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 firstly, y=√11= 3.316


Now, squaring y, we get, 

 

  y2= (3.316)2=11



 or, y2=11


Answer : 11

Explanation

 squaring “y” which is same as squaring the value of √11 resulted to11
 

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Problem 5

Find √11 / √9

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√11/√9

 

= √(11/9)

 

= 3.316/3

 

= 1.10533


Answer : 1.10533 
 

Explanation

dividing the square root value of 11 with that of square root value of 9
 

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FAQs on 11 Square Root

1.What is the √11 in fraction?

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2.Is √11 a real number?

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3.Is 11 a perfect square or non-perfect square?

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4.Is the square root of 11 a rational or irrational number?

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5.How would you represent √11 on a number line?

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6.What is the value of √7 ?

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7.How does learning Algebra help students in United States make better decisions in daily life?

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8.How can cultural or local activities in United States support learning Algebra topics such as Square root of 11?

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9.How do technology and digital tools in United States support learning Algebra and Square root of 11?

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10.Does learning Algebra support future career opportunities for students in United States?

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Professor Greenline from BrightChamps

Important Glossaries for Square Root of 11

  • Exponential form:  An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors. It includes continuous multiplication involving base and exponent. Ex: 2 × 2 × 2 × 2 = 16 Or, 2 4 = 16, where 2 is the base, 4 is the exponent 

 

  • Prime Factorization:  Expressing the given expression as a product of its factors. Ex: 48=2 × 2 × 2 × 2 × 3

 

  • Prime Numbers: Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,....

 

  • Rational numbers and Irrational numbers: The Number which can be expressed as p/q, where p and q are integers and q not equal to 0 are called Rational numbers. Numbers which cannot be expressed as p/q, where p and q are integers and q not equal to 0 are called Irrational numbers. 

 

  • Perfect and non-perfect square numbers: Perfect square numbers are those numbers whose square roots do not include decimal places. Ex: 4,9,25 Non-perfect square numbers are those numbers whose square roots comprise decimal places. Ex :3, 8, 24
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About BrightChamps in United States

At BrightChamps, we understand algebra is more than just symbols—it’s a gateway to endless possibilities! Our goal is to empower kids throughout the United States to master key math skills, like today’s topic on the Square root of 11, with a special emphasis on understanding square roots—in an engaging, fun, and easy-to-grasp manner. Whether your child is calculating how fast a roller coaster zooms through Disney World, keeping track of scores during a Little League game, or budgeting their allowance for the latest gadgets, mastering algebra boosts their confidence to tackle everyday problems. Our hands-on lessons make learning both accessible and exciting. Since kids in the USA learn in diverse ways, we customize our methods to suit each learner’s style. From the lively streets of New York City to the sunny beaches of California, BrightChamps brings math alive, making it meaningful and enjoyable all across America. Let’s make square roots an exciting part of every child’s math adventure!
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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

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