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.

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

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.

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.

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.

• 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.