Square Root of 10400
2026-02-21 20:44 Diff
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Last updated on August 5, 2025

If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in various fields such as vehicle design and finance. Here, we will discuss the square root of 10400.

What is the Square Root of 10400?

The square root is the inverse of the square of the number. 10400 is not a perfect square. The square root of 10400 is expressed in both radical and exponential form. In radical form, it is expressed as √10400, whereas it is expressed as (10400)^(1/2) in exponential form. √10400 = 102, which is a rational number because it can be expressed in the form of p/q, where p and q are integers and q ≠ 0.

Finding the Square Root of 10400

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, the long-division method and approximation method are used. Let us now learn the following methods:

  • Prime factorization method
  • Long division method
  • Approximation method

Square Root of 10400 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 10400 is broken down into its prime factors.

Step 1: Finding the prime factors of 10400.

Breaking it down, we get 2 x 2 x 2 x 2 x 5 x 5 x 13 x 20: 2^4 x 5^2 x 13 x 20.

Step 2: Now we found out the prime factors of 10400. The second step is to make pairs of those prime factors. Since 10400 is a perfect square, the digits of the number can be grouped in pairs. Therefore, calculating √10400 using prime factorization is possible.

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Square Root of 10400 by Long Division Method

The long division method is particularly used for non-perfect square numbers, but it can also be used for perfect squares. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, we need to group the numbers from right to left. In the case of 10400, we need to group it as 400 and 104.

Step 2: Now we need to find n whose square is closest to 104. We can say n is ‘10’ because 10 x 10 is less than or equal to 104. Now the quotient is 10, and after subtracting 100 from 104, the remainder is 4.

Step 3: Now let us bring down 00, which is the new dividend. Add the old divisor with the same number 10 + 10 to get 20, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 20n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 20n × n ≤ 400. Let us consider n as 2; now 20 x 2 x 2 = 400.

Step 6: Subtract 400 from 400, and the difference is 0.

Step 7: Since the remainder is zero, the quotient is 102.

So the square root of √10400 is 102.

Square Root of 10400 by Approximation Method

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 10400 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √10400. The smallest perfect square less than 10400 is 10000, and the largest perfect square greater than 10400 is 10404. √10400 falls somewhere between 100 and 102.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Going by the formula (10400 - 10000) ÷ (10404 - 10000) = 100/404 = 0.25

Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 100 + 0.25 = 102.5, so the square root of 10400 is approximately 102.

Common Mistakes and How to Avoid Them in the Square Root of 10400

Students do make mistakes while finding the square root, such as forgetting about the negative square root or skipping long division methods. Now let us look at a few of those mistakes that students tend to make in detail.

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Problem 1

Can you help Max find the area of a square box if its side length is given as √10400?

Okay, lets begin

The area of the square is 10400 square units.

Explanation

The area of the square = side².

The side length is given as √10400.

Area of the square = side² = √10400 x √10400 = 102 x 102 = 10400.

Therefore, the area of the square box is 10400 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 10400 square feet is built; if each of the sides is √10400, what will be the square feet of half of the building?

Okay, lets begin

5200 square feet.

Explanation

We can just divide the given area by 2 as the building is square-shaped.

Dividing 10400 by 2 gives us 5200.

So half of the building measures 5200 square feet.

Well explained 👍

Problem 3

Calculate √10400 x 5.

Okay, lets begin

510.

Explanation

The first step is to find the square root of 10400, which is 102.

The second step is to multiply 102 with 5.

So 102 x 5 = 510.

Well explained 👍

Problem 4

What will be the square root of (10000 + 400)?

Okay, lets begin

The square root is 102.

Explanation

To find the square root, we need to find the sum of (10000 + 400). 10000 + 400 = 10400, and then √10400 = 102.

Therefore, the square root of (10000 + 400) is ±102.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √10400 units and the width ‘w’ is 50 units.

Okay, lets begin

We find the perimeter of the rectangle as 304 units.

Explanation

Perimeter of the rectangle = 2 × (length + width).

Perimeter = 2 × (√10400 + 50) = 2 × (102 + 50) = 2 × 152 = 304 units.

Well explained 👍

FAQ on Square Root of 10400

1.What is √10400 in its simplest form?

The prime factorization of 10400 is 2^4 x 5^2 x 13 x 20, so the simplest form of √10400 = √(2^4 x 5^2 x 13 x 20) = 102.

2.Mention the factors of 10400.

Factors of 10400 include 1, 2, 4, 5, 8, 10, 13, 16, 20, 25, 26, 32, 40, 50, 52, 65, 80, 100, 104, 130, 160, 200, 208, 260, 325, 400, 416, 520, 650, 800, 1040, 1300, 2080, 2600, 5200, and 10400.

3.Calculate the square of 10400.

We get the square of 10400 by multiplying the number by itself, that is 10400 x 10400 = 108160000.

4.Is 10400 a prime number?

10400 is not a prime number, as it has more than two factors.

5.10400 is divisible by?

10400 has many factors, and it is divisible by numbers such as 1, 2, 4, 5, 8, 10, 13, 16, 20, 25, 26, 32, 40, 50, 52, 65, 80, 100, 104, 130, 160, 200, 208, 260, 325, 400, 416, 520, 650, 800, 1040, 1300, 2080, 2600, 5200, and 10400.

Important Glossaries for the Square Root of 10400

  • Square root: A square root is the inverse of a square. For example, 10^2 = 100, and the inverse of the square is the square root, which is √100 = 10.
     
  • Rational number: A rational number is a number that can be expressed in the form of p/q, where p and q are integers and q is not equal to zero.
     
  • Perfect square: A perfect square is a number that can be expressed as the square of an integer. For example, 100 is a perfect square as it can be expressed as 10^2.
     
  • Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors.
     
  • Long division method: The long division method is a technique used to find the square root of a number by dividing the number into groups and finding the square root using division and subtraction.

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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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