Square root of 3
2026-02-28 17:11 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-3

1316 Learners

Last updated on August 5, 2025

The square root of 3 is the inverse operation of squaring a value “y” such that when “y” is multiplied by itself → y ⤫ y, the result is 3. It contains both positive and a negative root, where the positive root is called the principal square root.

What Is the Square Root of 3?

The square root of 3 is ±1.73205080757.
The positive value, 1.73205080757 is the solution of the equation x2 = 3. As defined, the square root is just the inverse of squaring a number, so, squaring 1.73205080757 will result in 3.  The square root of 3 is expressed as √3 in radical form, where the ‘√’  sign is called “radical”  sign. In exponential form, it is written as (3)1/2  
 

Finding the Square Root of 3

We can find the square root of 3 through various methods. They are:

  • Prime factorization method
  • Approximation/Estimation method
     

Square Root of 3 By Prime Factorization Method

The prime factorization of 3 involves breaking down a number into its factors. Divide 3 by prime numbers, and continue to divide the quotients until they can’t be separated anymore. After factorizing 3, 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 3 = 3 × 1   


for 3, no pairs of factors can be obtained, only a single 3 is obtained.


So, it can be expressed as  √3 = √(3 × 1) = √3


√3 is the simplest radical form of √3.
 

Explore Our Programs

Square Root of 3 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 3:


Step 1: Write the number 3, 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 3. Here, it is 1, Because 12=1< 3.


Step 3 : Now divide 3 by 1 (the number we got from Step 2) such that we get 1 as quotient and we get a remainder. Double the divisor 1, we get 2, and then the largest possible number A1=7 is chosen such that when 7 is written beside the new divisor, 2, a 2-digit number is formed →27, and multiplying 7 with 27 gives 189 which is less than 200.

Repeat the process until you reach the remainder of 0

We are left with the remainder, 176 (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 1.732….

Square Root of 3 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 3


Below : 1→ square root of 1 = 1     ……..(i)


Above : 4 →square root of 4 = 2     ……..(ii)


Step 2 : Divide 3 with one of 1 or 2


 If we choose 2, and divide 3 by 2, we get 1.5   …….(iii)

              Step 3: Find the average of 2 (from (ii)) and 1.5 (from (iii))


(2+1.5)/2 = 1.75


 Hence, 1.75 is the approximate square root of 3
 

Common Mistakes and How to Avoid Them in the Square Root of 3

When we find the square root of 3, 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 √3 + 5√3 ?

Okay, lets begin

√3 + 5√3

= √3(1+5)

= 6√3

Answer : 6√3

Explanation

the simplest radical form of √3 is √3, so, we applied that and solved.
 

Well explained 👍

Problem 2

What is √3 multiplied by 2√3 and then divided by (√3)²?

Okay, lets begin

 (√3 ⤬ 2√3)/(√3)2

= (3⤬2)/3

= 2


Answer: 2 
 

Explanation

√3  multiplying with itself gives 3, and then again multiplied by 2 in the first step and then again divided by (√3)2=3 
 

Well explained 👍

Problem 3

Find the value of 1/√3?

Okay, lets begin

1/√3

= 1/ 1.73

=0.578


Answer: 0.578
 

Explanation

we divide 1 by the value of √3
 

Well explained 👍

Problem 4

If y=√3, find y², y^3, y^4

Okay, lets begin

firstly, y=√3 


Now, squaring y, we get,


y2= (√3)2=3


or, y2=3


Similarly,  y3=(√3)3=3√3


Similarly, y4=(√3)4=((√3)2)2=9


Answer : 3, 3√3, 9
 

Explanation

 squaring “y” which is same as squaring the value of √3 resulted to 3 and hence applied this fact to each problem here.
 

Well explained 👍

Problem 5

Find √3 / √3

Okay, lets begin

√3/√3

= √(3/3)

= √1

= 1


Answer : 1 
 

Explanation

since the numerator and denominator is same, the answer is 1
 

Well explained 👍

FAQs on 3 Square Root

1.What is the value of √3?

The value of √3 is 1.73205080757.

2.What is the value of 3√2?

 The value of 3√2 = 3  ╳ √2= 3╳ 1.4142135627 =4.242
 

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

3 is a non-perfect square, since 3 =(1.73205080757)2.
 

4.Is the square root of 3 a rational or irrational number?

The square root of 3 is ±1.73205080757. So, 1.73205080757 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.Is 3 a perfect cube?

Important Glossaries for Square Root of 3

  • 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.