Square root of 5
2026-02-28 10:30 Diff
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Last updated on August 5, 2025

The square root of 5 is the inverse operation of squaring a value “y” such that when “y” is multiplied by itself → y × y, the result is 5. It contains both positive and a negative root, where the positive root is called the principal square root.

What Is the Square Root of 5?

The square root of 5 is ±2.2360. The positive value, 2.2360 is the solution of the equation x2 = 5. As defined, the square root is just the inverse of squaring a number, so, squaring 2.2360 will result in 5. The square root of 5 is expressed as √5 in radical form, where the ‘√’  sign is called “radical” sign. In exponential form, it is written as (5)1/2  
 

Finding the Square Root of 5

We can find the square root of 5 through various methods. They are:

  • Prime factorization method
  • Approximation/Estimation method

Square Root of 5 By Prime Factorization Method

The prime factorization of 5 involves breaking down a number into its factors. Divide 5 by prime numbers, and continue to divide the quotients until they can’t be separated anymore. After factorizing 5, 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 5 = 5 × 1

  
for 5, no pairs of factors can be obtained, but a single 5 is obtained.


So, it can be expressed as  √5 = √(5 × 1) = √5


√5 is the simplest radical form of √5

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Square Root of 5 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 5:


Step 1: Write the number 5, 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 5. Here, it is 2, Because 22=4 < 5.


Step 3 : Now divide 5 by 2 (the number we got from Step 2) such that we get 2 as quotient and we get a remainder.  Double the divisor 2, we get 4, and then the largest possible number A1=2 is chosen such that when 2 is written beside the new divisor, 4, a 2-digit number is formed →42, and multiplying 2 with 42 gives 84 which is less than 100.

Repeat the process until you reach the remainder of 0


We are left with the remainder, 27100 (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 2.23….

Square Root of 5 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 5


Below : 4→ square root of 4 = 2     ……..(i)


Above : 9 →square root of 9 = 3     ……..(ii)

Step 2 : Divide 5 with one of 2 or 3

If we choose 2, and divide 5 by 2, we get 2.5   …….(iii)

              Step 3: Find the average of 2 (from (i)) and 2.5 (from (iii))


(2+2.5)/2 = 2.25

            
Hence, 2.25 is the approximate square root of 5
 

Common Mistakes and How to Avoid Them in the Square Root of 5

When we find the square root of 5, 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 + 5√5 + 10√5 ?

Okay, lets begin

 √5 + 5√5+10√5

= √5(1+5+10)

= 16√5


Answer : 16√5
 

Explanation

The simplest radical form of √5 is √5, so, it is taken common outside and calculated simply.
 

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Problem 2

What is √5 multiplied by 5√5?

Okay, lets begin

√5 ⤬ 5√5

= 5⤬5

= 25


Answer: 25
 

Explanation

 √5  multiplying with itself gives 5, and then again multiplied by 5 
 

Well explained 👍

Problem 3

Find the value of 1/√5?

Okay, lets begin

1/√ 5

= 1/ 2.23

=0.4291


Answer: 0.4291
 

Explanation

we divide 1 by the value of √5
 

Well explained 👍

Problem 4

If y=√5, find y²

Okay, lets begin

firstly, y=√5

= 2.2360


 Now, squaring y, we get, 


y2= (2.2360)2

=5


or, y2=5


Answer : 5
 

Explanation

 squaring “y” which is same as squaring the value of √5 resulted to 5
 

Well explained 👍

Problem 5

Approximate √5 up to three decimal places.

Okay, lets begin

√5 ≅ 2.236 (up to three decimal places)


Answer : 2.236 
 

Explanation

 finding the square root of 5 up till three decimal places.
 

Well explained 👍

FAQs on 5 Square Root

1.Is √5 a real number?

2.Is √5 a whole number ?

3.Is 5 a perfect square or non-perfect square?

 5 is a non-perfect square, since 5 =(2.2360) 2.
 

4.Is the square root of 5 a rational or irrational number?

The square root of 5 is ±2.2360. So, 2.2360 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. How to solve √10?

√10 can be solved by methods to find square roots:


Long Division Method, Prime Factorization, Approximation method. The value of √10 is 3.162277
 

Important Glossaries for Square Root of 5

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

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