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  
 

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

• Prime factorization method

• Long division method

• Approximation/Estimation 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

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…

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

• 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