Square Root of 250
2026-02-28 13:12 Diff
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Last updated on August 5, 2025

A square root is a number that, when we double it, it gives you another number. It is a very important and interesting part of mathematics. You must have applied it for measuring each side of a square from the total area.

What is the Square Root of 250?

The square root of 250 is a number, when we multiply it by itself we get 250. The square root of 250 is an irrational number. As it cannot be written as a ratio of two numbers. It is denoted by 250 and is approximately equal to 15.8114. 


Exponential form : 2501/2  ≅ 15.8114.


Radical Form: √250 

Finding the Square Root of 250

We can find the square root of a number by using methods like: Prime Factorization; Long Division method; Approximation method and Subtraction method:
 

Prime Factorization

The factoring of a number into smaller numbers is prime factorization. Here, 250 is a composite number, it can be broken down into smaller numbers more than 2.


250 = 2×5×5×5 = 2×53


Taking out the perfect square,


√250  = √25x10 = 5√10


 So, from this method we cannot find the exact square root, but we confirm that 250 is not a perfect square.

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Long Division Method

In this method, we get to find the value of the square root precisely.

Grouping the digits: We start with pairing the digits from the decimal part 250.00


Find the number whose square will be less than or equal to 250 i.e., 152=225


Subtract 152=225 from 250, which leaves us with 25.


Now we bring down two zeros, which makes it 2500


Next double the divisor 15, we get 30. Next we find the largest digit which will be lesser than or equal to 2500.


Repeat the steps to get the next decimal places.


So after calculation we get, √250  = 15.8114

Approximation Method

As 152 =225 and 162 =256, the square root of 250 lies between 15 and 16.


Start by guessing 15.62 which is nearest to 15.


15.82  = 249.64 which too less


Go to the next number 15.7, 15.822 = 250.27 which is too high.


So, 250  = 15.811
 

Subtraction Method

 The subtraction method includes subtracting consecutive odd numbers from 250 to see how many steps we need to reach zero. However, since 250 is not a perfect square, we cannot exactly reach 0. 

250 -1 =249


249-3=246


246-5=241

As we did not get zero, we understand that 250 is not a perfect square.
 

Common Mistakes and How to Avoid Them in the Square Root of 250

While learning about Square root, students are likely to make mistakes. Below-mentioned are a few mistakes with solutions on how to avoid them:
 

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Problem 1

If, x²=250, find the value of x.

Okay, lets begin

If, x2=250

Then, 


x= 250 
 

Explanation

 Here is the square when shifted to the RHS it becomes the square root of the number


x= 577 


x=75


So the value of x is 75.

Well explained 👍

Problem 2

Verify if √250 is greater than 15.

Okay, lets begin

First, approximate √250 :


Using prime factorization:


√250 = √5x49 = 7√5


Since √5 = 2.236


7√5 = 7 × 2.236 =15.652


Since 15.652>15, we conclude that : 


√250 > 15

Explanation

The approximation of  √250 shows us that it is greater than 15.

Well explained 👍

Problem 3

Express the square root of 250 in the simplest radical form.

Okay, lets begin

√250  = 5 × 72


We use the square root property:


√250 =√5x7x7 


Now take out the square of the number which is 7×7 = 49, take out 7.


√250 = 7√5 


So, the value is  7√5 .

Explanation

Square root in terms of radical form is  7√5

Well explained 👍

Problem 4

Solve: 20/√250

Okay, lets begin

 To simplify, 20/ √250 , we multiply the number in the denominator with the numerator and the denominator, which is called rationalizing. 

Explanation

20/√250  x   √250 / √250  = 20√250 /√250  , here when two square roots with the same number are multiplied the roots get canceled (in the denominator), and we are left with the same number, hence√ 250 x √250  = 250.


After rationalizing, we get,  20 √250 / 250 

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Problem 5

Simplify √25+√250

Okay, lets begin

√25+√250     here we have a root inside a root, which is called a nested root. 


Explanation: We first find the square root of the number that is inside,   √250 =15.811


=√ 25+15.811 


Now we add both the numbers. We get,


= √40.811   , next we find the square root of 40.811


= 6.39

Explanation

Answer:    = 6.39
 

Well explained 👍

FAQs on 250 Square Root

1.Is 250 a perfect square root?

250 is not a perfect square number, as we cannot find any number that can be multiplied to get 250.
 

2.What are the factors of 250?

The factors of 250 are 1,2,5,10,25,50,125 and 250. It is a composite number.
 

3.What is the value of √250 the nearest whole number?

4.What perfect Squares go into 250?

Solution: 25×5×2=250


The greatest perfect square which is the factor of 250 and is required is 25.
 

5.What is 250 a perfect cube?

In a triplet, the prime factor 2 will not appear. Hence, 250 is not a perfect cube.
 

Important Glossaries for Square Root of 250

  • Irrational number: A number that cannot be written in the form of a ratio or fraction. For example, √250 is an irrational number.
  • Exponent form: Writing the square root of a number in the form of degree or powers. For example, 2501/2  ≅ 4.7958
  • Square root: It is a number that, when we double it, it gives you another number. For example, 4 × 4 =16.
     

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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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