Square Root of 10368

The square root is the inverse of the square of the number. 10368 is not a perfect square. The square root of 10368 is expressed in both radical and exponential form. In the radical form, it is expressed as √10368, whereas (10368)^(1/2) in the exponential form. √10368 ≈ 101.823376, which is an irrational number because it cannot 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, the prime factorization method is not typically used for non-perfect square numbers where 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 10368 is broken down into its prime factors.

Step 1: Finding the prime factors of 10368.

Breaking it down, we get 2^6 × 3^4: 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3.

Step 2: Now we found out the prime factors of 10368. The second step is to make pairs of those prime factors. Since 10368 is not a perfect square, therefore the digits of the number can’t be grouped into a complete pair. Therefore, calculating 10368 using prime factorization directly is not feasible.

The long division method is particularly used for non-perfect square numbers. 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 10368, we group it as 10 and 368.

Step 2: Now we need to find n whose square is close to 10. We can say n is ‘3’ because 3 x 3 = 9 is lesser than or equal to 10. Now the quotient is 3, after subtracting 9 from 10, the remainder is 1.

Step 3: Now let us bring down 368, which is the new dividend. Add the old divisor with the same number, 3 + 3, we get 6, which will be part of our new divisor.

Step 4: The new divisor will be in the form 6n. We need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 1368. Let us consider n as 2, now 62 x 2 = 124.

Step 6: Subtract 124 from 1368, the difference is 144, and the quotient is 32.

Step 7: Since there are more digits left, bring them down and continue with the division process.

Step 8: Add a decimal point and continue the division until two decimal places are achieved.

So the square root of √10368 ≈ 101.82.

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 10368 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √10368. The smallest perfect square less than 10368 is 10000, and the largest perfect square greater than 10368 is 10404. √10368 falls somewhere between 100 and 102.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). (10368 - 10000) / (10404 - 10000) = 368 / 404 ≈ 0.91188.

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 + 0.91188 ≈ 100.91188, so the square root of 10368 is approximately 101.82.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, so √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be expressed as a simple fraction; it's non-repeating and non-terminating when written as a decimal.
   
• Principal square root: The principal square root is the non-negative square root of a number. Although a number has both positive and negative square roots, the principal square root is generally used.
   
• Decimal: A decimal is a number that is expressed in the base-10 numeral system, which consists of a whole number part and a fractional part separated by a decimal point.
   
• Long division method: A technique used to find the square root of a number by systematically dividing the number and finding its closest approximate root.