Square root of 52
2026-02-28 19:15 Diff
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Last updated on August 5, 2025

The square root of 52 is the inverse operation of squaring a value “y” such that when “y” is multiplied by itself → y × y, the result is 52. It contains both positive and a negative root, where the positive root is called the principal square root.

What Is the Square Root of 52?

The square root of 52 is ±7.21110255093. The positive value,7.21110255093 is the solution of the equation x2 = 52. As defined, the square root is just the inverse of squaring a number, so, squaring 7.21110255093 will result in 52. The square root of 52 is expressed as √52 in radical form, where the ‘√’  sign is called “radical”  sign. In exponential form, it is written as (52)1/2  
 

Finding the Square Root of 52

We can find the square root of 52 through various methods. They are:

  • Prime factorization method
  • Approximation/Estimation method

Square Root of 52 By Prime Factorization Method


The prime factorization of 52 involves breaking down a number into its factors. Divide 52 by prime numbers, and continue to divide the quotients until they can’t be separated anymore. After factoring 52, 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 52 = 13 × 2 ×2   


for 52, one pairs of factors 2 obtained, but a single 13 is also obtained.


So, it can be expressed as  √52 = √(2 × 2  × 13) = 2√13


2√13 is the simplest radical form of √52

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Square Root of 52 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 52:


Step 1 : Write the number 52, 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 52. Here, it is 7, Because 72=49 < 52

Step 3 : Now divide 52 by 7 (the number we got from Step 2) such that we get 7 as quotient, and we get a remainder. Double the divisor 7, we get 14 and then the largest possible number A1=2 is chosen such that when 2 is written beside the new divisor, 14, a 3-digit number is formed →142 and multiplying 2 with 142 gives 284 which is less than 300.

Repeat the process until you reach remainder 0


We are left with the remainder, 1479 (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 7.211…

Square Root of 52 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 52


Below : 49→ square root of 49 = 7     ……..(i)


Above : 64 →square root of 64= 8     ……..(ii)


Step 2 : Divide 52 with one of 7 or 8 


If we choose 7, and divide 52 by 7, we get 7.428   …….(iii)

              Step 3: Find the average of 7 (from (i)) and 7.428 (from (iii))


(7+7.428)/2 = 7.2142

            
 Hence, 7.2142 is the approximate square root of 52

Common Mistakes and How to Avoid Them in the Square Root of 52

When we find the square root of 52, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.
 

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Problem 1

Simplify 5√52 + 13√52 ?

Okay, lets begin

 5√52+13√52

= √52(5+13)

= 7.211 ⤬ (5+13)

=18 ⤬ 7.211

= 129.7998 


Answer : 129.7998
 

Explanation

Taking out the common part √52, adding the values inside the bracket. √52= 7.2111, so multiplying the square root value with the sum
 

Well explained 👍

Problem 2

√52 with 13

Okay, lets begin

√52  ⤬ 13

= 2√13⤬ 13

=26√13


Answer : 26√13
 

Explanation

multiplying the simplest radical form of √52 with 13.
 

Well explained 👍

Problem 3

Compare √52 and √48

Okay, lets begin

 √52 ≅ 7.2111,

√48 ≅ 6.928


So, √52 is greater than √48


Answer: √52 > √48
 

Explanation

 finding out the approximate values of √52 and √48 and comparing them

Well explained 👍

Problem 4

If y=√52, find y²

Okay, lets begin

 firstly, y=√52= 7.21110255093


Now, squaring y, we get, 


y2= (7.21110255093)2=52


or, y2=52


Answer : 52
 

Explanation

squaring “y” which is same as squaring the value of √52 resulted to  52
 

Well explained 👍

Problem 5

Find √52 / √48

Okay, lets begin

√52/√48

= √(52/49)

= 7.2111 / 6.928

= 1.04086316


Answer : 1.04086316 
 

Explanation

 dividing the square root value of 52 with that of square root value of 48
 

Well explained 👍

FAQs on Square Root of 52

1.How to solve √50?

√50 can be solved through methods like Long Division Method, Prime Factorization method, Approximation method 
 

2.What is 52 divisible by ?

Factors of 52 are: 1,2,4,13,26, and 52
 

3.Is 52 a perfect square or non-perfect square?

52 is a non-perfect square, since 52 =(7.21110255093) 2.
 

4.Is the square root of 52 a rational or irrational number?

The square root of 52 is ±7.21110255093. So, 7.21110255093 is an irrational number, since it cannot be obtained by dividing two integers and cannot be written in the form p/q, where p and q are integers.
 

5. 52 falls between which two perfect squares?

: 52 falls between perfect squares → 49 and 64
 

6. What is the closest perfect square to 52?

49 is the closest perfect square to 52
 

Important Glossaries for Square Root of 52

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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Jaskaran Singh Saluja

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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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