Square Root of 1392

The square root is the inverse of the square of a number. 1392 is not a perfect square. The square root of 1392 is expressed in both radical and exponential form. In the radical form, it is expressed as √1392, whereas (1392)^(1/2) in the exponential form. √1392 ≈ 37.282, 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 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 1392 is broken down into its prime factors.

Step 1: Finding the prime factors of 1392.

Breaking it down, we get 2 x 2 x 2 x 2 x 3 x 29: 2^4 x 3^1 x 29^1.

Step 2: Now we found out the prime factors of 1392. The second step is to make pairs of those prime factors. Since 1392 is not a perfect square, therefore the digits of the number can’t be grouped in pairs completely. Therefore, calculating 1392 using prime factorization is not straightforward.

The long division method is particularly used for non-perfect square numbers. 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 1392, we need to group it as 92 and 13.

Step 2: Now we need to find n whose square is 13. We can say n as ‘3’ because 3 x 3 = 9, which is less than 13. Now the quotient is 3, after subtracting 9 from 13, the remainder is 4.

Step 3: Now let us bring down 92, forming a new dividend of 492. Add the old divisor with the same number 3 + 3 = 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 492. Let us consider n as 8, now 68 x 8 = 544, which is too large.

Step 6: Try n as 7, so 67 x 7 = 469.

Step 7: Subtract 469 from 492, the difference is 23, and the quotient is 37.

Step 8: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 2300.

Step 9: We need to find the new divisor, using 746 x 3 = 2238.

Step 10: Subtract 2238 from 2300, and we get 62.

Step 11: The current quotient is 37.3.

Step 12: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal, continue till the remainder is zero.

So the square root of √1392 is approximately 37.28.

Approximation method is another method for finding the 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 1392 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1392. The smallest perfect square less than 1392 is 1369 (37^2), and the largest perfect square greater than 1392 is 1444 (38^2). √1392 falls somewhere between 37 and 38.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (1392 - 1369) ÷ (1444 - 1369) = 23 / 75 ≈ 0.3067. 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 37 + 0.3067 ≈ 37.31, so the square root of 1392 is approximately 37.31.

• 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, that is, √16 = 4.

• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.

• Principal square root: A number has both positive and negative square roots; however, it is the positive square root that is typically used in real-world applications, known as the principal square root.

• Prime factorization: Breaking down a number into the set of prime numbers that multiply together to result in the original number.

• Decimal: A number that contains a whole number and a fractional part separated by a decimal point, such as 7.86, 8.65, and 9.42.