Square Root of 1412

The square root is the inverse of the square of the number. 1412 is not a perfect square. The square root of 1412 is expressed in both radical and exponential form. In the radical form, it is expressed as √1412, whereas (1412)^(1/2) in the exponential form. √1412 ≈ 37.558, 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 1412 is broken down into its prime factors:

Step 1: Finding the prime factors of 1412 Breaking it down, we get 2 × 2 × 353: 2^2 × 353

Step 2: Now, we have found the prime factors of 1412. The second step is to make pairs of those prime factors. Since 1412 is not a perfect square, therefore, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1412 using prime factorization is impossible.

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 1412, we need to group it as 12 and 14.

Step 2: Now, we need to find n whose square is less than or equal to 14. We can say n as ‘3’ because 3 × 3 = 9 is lesser than 14. Now the quotient is 3. After subtracting 9 from 14, the remainder is 5.

Step 3: Now, let us bring down 12 which is the new dividend. Add the old divisor with the same number 3 + 3, we get 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, and we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 512. Let us consider n as 8. Now, 68 × 8 = 544

Step 6: Since 544 is greater than 512, we consider n as 7. Therefore, 67 × 7 = 469

Step 7: Subtract 469 from 512; the difference is 43. 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 4300.

Step 9: Now, we need to find the new divisor, which is 754 because 754 × 4 = 3016

Step 10: Subtracting 3016 from 4300, we get the result 1284.

Step 11: The quotient is now 37.4. Step 12: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue until the remainder is zero.

So, the square root of √1412 is approximately 37.558.

The 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 1412 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1412.

The smallest perfect square closest to 1412 is 1369, and the largest perfect square is 1444. √1412 falls somewhere between 37 and 38.

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 (1412 - 1369) ÷ (1444 - 1369) = 43 ÷ 75 ≈ 0.573

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 + 0.573 = 37.573.

So, the square root of 1412 is approximately 37.573.

• Square root: A square root is the inverse of a square. For 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.

• Long division method: A mathematical method used to divide large numbers and find square roots of non-perfect squares through a series of steps.

• Approximation method: A technique for estimating the value of the square root by finding the nearest perfect squares and calculating a decimal approximation.

• Prime factorization: Breaking down a composite number into a product of its prime factors.