Square Root of 1377

The square root is the inverse of the square of the number. 1377 is not a perfect square. The square root of 1377 is expressed in both radical and exponential form. In the radical form, it is expressed as √1377, whereas (1377)^(1/2) in the exponential form. √1377 ≈ 37.104, 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 1377 is broken down into its prime factors.

Step 1: Finding the prime factors of 1377

Breaking it down, we get 3 x 3 x 3 x 3 x 17: 3^4 x 17

Step 2: Now we found out the prime factors of 1377. The second step is to make pairs of those prime factors. Since 1377 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 1377 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 1377, we need to group it as 77 and 13.

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

Step 3: Now let us bring down 77 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, we need to find the value of n.

Step 5: The next step is finding 6n x n ≤ 477. Let us consider n as 7, now 6 x 7 x 7 = 294.

Step 6: Subtract 477 from 294, the difference is 183, and the quotient is 37.

Step 7: 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 18300.

Step 8: Now we need to find the new divisor that is 741 because 741 x 1 = 741.

Step 9: Subtracting 741 from 18300, we get the result 17559.

Step 10: Now the quotient is 37.1.

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

So the square root of √1377 is approximately 37.10.

The approximation method is another method for finding the square roots, and 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 1377 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1377. The smallest perfect square less than 1377 is 1369 and the largest perfect square greater than 1377 is 1444. √1377 falls somewhere between 37 and 38.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Largest perfect square - smallest perfect square). Going by the formula (1377 - 1369) ÷ (1444 - 1369) = 8/75 = 0.107. 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 + 0.107 = 37.107, so the square root of 1377 is approximately 37.107.

• 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 usually the positive square root that is used more often in real-world situations. That is the reason it is also known as the principal square root.

• Prime factorization: It is the process of expressing a number as a product of its prime factors. For example, the prime factorization of 18 is 2 x 3 x 3.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.