Square Root of 1368

The square root is the inverse of the square of a number. 1368 is not a perfect square. The square root of 1368 is expressed in both radical and exponential form. In the radical form, it is expressed as √1368, whereas (1368)^(1/2) in the exponential form. √1368 ≈ 36.9858, 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 used for non-perfect square numbers, where 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 1368 is broken down into its prime factors.

Step 1: Finding the prime factors of 1368.

Breaking it down, we get 2 × 2 × 2 × 3 × 3 × 19: 2^3 × 3^2 × 19

Step 2: Now that we have found the prime factors of 1368, the second step is to make pairs of those prime factors. Since 1368 is not a perfect square, the digits of the number can’t be fully grouped in pairs. Therefore, calculating 1368 using prime factorization is limited in providing an exact integer result.

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 1368, we need to group it as 68 and 13.

Step 2: Now we need to find n whose square is less than or equal to 13. We can say n is '3' because 3 × 3 = 9 is less than or equal to 13. Now the quotient is 3, after subtracting 9 from 13, the remainder is 4.

Step 3: Bring down 68, making the new dividend 468. Add the old divisor with the same number: 3 + 3 = 6, which will be our new divisor.

Step 4: The new divisor is 6n. We need to find n such that 6n × n ≤ 468. Let n be 7, then 67 × 7 = 469 which is slightly more than 468. Let's use n = 6, 66 × 6 = 396.

Step 5: Subtract 396 from 468, the difference is 72, and the quotient is 36.

Step 6: Add a decimal point to continue the division and bring down two zeros to the remainder, making it 7200.

Step 7: Find the next divisor. Use the quotient 369, so 369n ≤ 7200. Find suitable n; let's say n = 1, 369 × 1 = 369.

Step 8: Continue the division until the desired accuracy is reached. So the square root of √1368 is approximately 36.98.

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 1368 using the approximation method.

Step 1: Find the closest perfect squares around √1368. The smallest perfect square less than 1368 is 1296 (36^2) and the largest perfect square more than 1368 is 1444 (38^2). √1368 falls between 36 and 38.

Step 2: Apply the formula: (Given number - smaller perfect square) ÷ (larger perfect square - smaller perfect square). Using the formula: (1368 - 1296) ÷ (1444 - 1296) = 72 ÷ 148 ≈ 0.486 Add this decimal to the lower integer square root: 36 + 0.486 = 36.486, so the square root of 1368 is approximately 36.49.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. Example: 6^2 = 36, and the square root is √36 = 6.

• Irrational number: An irrational number is a number that cannot be expressed as a simple fraction; its decimal goes on forever without repeating.

• Approximation: The process of finding a value that is close enough to the correct answer, usually within a specified tolerance.

• Perfect square: A number that is the square of an integer. Example: 36 is a perfect square because it is 6^2.

• Prime factorization: Expressing a number as the product of its prime factors. Example: The prime factorization of 1368 is 2^3 × 3^2 × 19.