Square root of 200
2026-02-28 23:58 Diff
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Last updated on August 5, 2025

The square root of 200 is the inverse operation of squaring a value “y” such that when “y” is multiplied by itself → y × y, the result is 200. It contains both positive and a negative root, where the positive root is called the principal square root.

What Is the Square Root of 200?

The square root of 200 is ±14.1421356237.

The positive value, 14.1421356237 is the solution of the equation x2 = 200. As defined, the square root is just the inverse of squaring a number, so, squaring 14.1421356237 will result in 200.  The square root of 200 is expressed as √200 in radical form, where the ‘√’  sign is called “radical”  sign. In exponential form, it is written as (200)1/2  
 

Finding the Square Root of 200

We can find the square root of 200 through various methods. They are:

  • Prime factorization method
  • Approximation/Estimation method

Square Root of 200 By Prime Factorization Method

The prime factorization of 200 involves breaking down a number into its factors. Divide 200 by prime numbers, and continue to divide the quotients until they can’t be separated anymore.

After factorizing 200, 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 200 = 2  ×2 ×2 ×5 ×5  

  
for 200, two pairs of factors 2 and of factor 5 are obtained, but a single 2 is remaining.


So, it can be expressed as  √200 = √(2 ×2 ×2 × 5 ×5) = 10√2


10√2 is the simplest radical form of √200

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Square Root of 200 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 200:


Step 1 : Write the number 200, 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 2. Here, it is 1, Because 12=1 ≤1 

Step 3 : Now divide 200 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=4 is chosen such that when 4 is written beside the new divisor, 2, a 2-digit number is formed →24 and multiplying 4 with 24 gives 96 which is less than 100.

Repeat the process until you reach remainder 0


We are left with the remainder, 3836 (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 14.142…

Square Root of 200 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 200


Below : 196→ square root of 196 = 14     ……..(i)


 Above : 225 →square root of 225= 15     ……..(ii)


Step 2 : Divide 200 with one of 14 or 15


 If we choose 14, and divide 200 by 14, we get 14.2857   …….(iii)


             
Step 3: Find the average of 14 (from (i)) and 14.2857 (from (iii))


(14+14.2857)/2 = 14.14285…

            
 Hence, 14.1428 is the approximate square root of 200
 

Common Mistakes and How to Avoid Them in the Square Root of 200

When we find the square root of 200, 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 20√200?

Okay, lets begin

20√200

= 20⤬√200

= 20⤬ 14.142

= 282.84


Answer : 282.84
 

Explanation

 √200= 14.142, so multiplying the square root value with 20
 

Well explained 👍

Problem 2

What is (√200 + √100) ⤬√200 ?

Okay, lets begin

(√200+ √100) ⤬ √200

= (14.142+10)⤬14.142

= 24.142 ⤬ 14.142

=341.416


Answer: 341.416
 

Explanation

adding the square root value of 200 with that of, 100 and then multiplying the square root value of 200 with the sum.
 

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Problem 3

Find the value of (1/√200)⤬ (1/√200) ?

Okay, lets begin

 (1/√200)⤬ (1/√200)

= 1/200

= 0.005


Answer: 0.005
 

Explanation

we know, √200⤬√200 = 200 and then solved by dividing 1 by 200
 

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Problem 4

Find the difference between (√200)² - (√199)²

Okay, lets begin

 (√200)2 - (√199)2


= 200 -199


=1


Answer: 1
 

Explanation

find out the square values of √200 and √199 and then found the difference
 

Well explained 👍

Problem 5

Find √200 / √9

Okay, lets begin

 √200/√9

= √(200/9)

= 14.142/3

= 4.714


Answer : 4.714
 

Explanation

dividing the square root value of 200 with that of square root value of 9.

Well explained 👍

FAQs on Square Root of 200

1.What is the value of √5?

The square root value of √5 = ±2.2360679775
 

2.How to solve √250?

√250 can be solved through various methods like Long Division Method, Prime Factorization Method or Approximation Method. The value of √250 is 15.8113883008 
 

3.Is 200 a perfect square or non-perfect square?

 200 is a non-perfect square, since 200 =(14.1421356237)2.
 

4.Is the square root of 200 a rational or irrational number?

The square root of 200 is ±14.1421356237. So, 14.1421356237 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.Which perfect squares lie between 200 and 250?

225 is the only perfect square that lies between 200 and 250.
 

Important Glossaries for Square Root of 200

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
 

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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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