Square root of 21
2026-02-28 01:08 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-21

936 Learners

Last updated on August 5, 2025

The square root of 21 is the inverse operation of squaring a value “y” such that when “y” is multiplied by itself → y ⤫ y, the result is 21. It contains both positive and a negative root, where the positive root is called the principal square root.

What Is the Square Root of 21?

The square root of 21 is ±4.58257569496.


The positive value, 4.58257569496 is the solution of the equation x2 = 21. As defined, the square root is just the inverse of squaring a number, so, squaring 4.58257569496 will result in 21.  The square root of 21 is expressed as √21 in radical form, where the ‘√’  sign is called “radical”  sign. In exponential form, it is written as (21)1/2  

Finding the Square Root of 21

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

  • Prime factorization method
  • Approximation/Estimation method
     

Square Root of 21 By Prime Factorization Method

The prime factorization of 21 involves breaking down a number into its factors. Divide 21 by prime numbers, and continue to divide the quotients until they can’t be separated anymore. After factorizing 21, 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 21 = 3 × 7    


for 21, no pairs of factors are obtained, but a single 3 and a single 7 are obtained.


So, it can be expressed as  √21 = √(3 × 7) = √21


√21 is the simplest radical form of √21

Explore Our Programs

Square Root of 21 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 21:


Step 1 : Write the number 21, 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 21. Here, it is 4, Because 42=16 < 21

Step 3 : Now divide 21 by 4 (the number we got from Step 2) such that we get 4 as quotient, and we get a remainder. Double the divisor 4, we get 8 and then the largest possible number A1=5 is chosen such that when 5 is written beside the new divisor, 8, a 2-digit number is formed →85 and multiplying 5 with 85 gives 425 which is less than 500.

Repeat the process until you reach remainder 0


We are left with the remainder, 5276 (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 4.582…

Square Root of 21 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 21


Below : 16→ square root of 16 = 4     ……..(i)


 Above : 25 →square root of 25= 5     ……..(ii)


Step 2 : Divide 21 with one of 4 or 5

 If we choose 5, and divide 21 by 5, we get 4.2   …….(iii)

              Step 3: Find the average of 5 (from (ii)) and 4.2 (from (iii))


(5+4.2)/2 = 4.6

            
 Hence, 4.6 is the approximate square root of 21
 

Common Mistakes and How to Avoid Them in the Square Root of 21

When we find the square root of 21, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.

Download Worksheets

Problem 1

Simplify 7⤬3√21?

Okay, lets begin

7⤬3√21  

= 7⤬3⤬ 4.582

= 96.222


Answer : 96.222
 

Explanation

√21= 4.582, so multiplying the square root value with 7⤬3
 

Well explained 👍

Problem 2

What is (√21 + √25) ⤬√21 ?

Okay, lets begin

(√21+ √25) ⤬ √21

= (4.582+ 5)⤬4.582

= 9.582 ⤬ 4.582

=43.9047


Answer: 43.9047
 

Explanation

adding the square root value of 21 with that of 25 and then multiplying the square root value of 21 with the sum.
 

Well explained 👍

Problem 3

Find the value of (1/√21)⤬ (1/√21) ?

Okay, lets begin

(1/√21)⤬ (1/√21)

= 1/21

= 0.04761… 


Answer: 0.04761…
 

Explanation

we know, √21⤬√21 = 21 and then solved by dividing 1 by 21
 

Well explained 👍

Problem 4

Find the difference between (√21)² - (√20)²

Okay, lets begin

  (√21)2 - (√20)2


= 21 -20


=1


Answer: 1
 

Explanation

find out the square values of √21 and √20 and then found the difference
 

Well explained 👍

Problem 5

Find √21 / √9

Okay, lets begin

√21/√9

= √(21/9)

= 4.582/3

= 1.5273…


Answer : 1.5273…
 

Explanation

dividing the square root value of 21 with that of square root value of 9
 

Well explained 👍

FAQs on Square Root of 21

1.How to solve √17?

√17 can be solved through various methods like Long Division Method, Prime Factorization method or Approximation Method. The value of √17 is 4.1231… 
 

2.How to solve √33?

√33 can be solved through various methods like Long Division Method, Prime Factorization method or Approximation Method. The value of √33 is 5.7445… 

3.Is 21 a perfect square or non-perfect square?

21 is a non-perfect square, since 21 =(4.58257569496)2.
 

4.Is the square root of 21 a rational or irrational number?

The square root of 21 is ±4.58257569496. So, 4.58257569496 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.What are the factors of 21?

 Factors of 21 are 1,3,7 and 21.
 

Important Glossaries for Square Root of 21

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, 24 = 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

What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math

▶

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.

Fun Fact

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