Square Root of 1384

The square root is the inverse of the square of the number. 1384 is not a perfect square. The square root of 1384 is expressed in both radical and exponential form. In the radical form, it is expressed as √1384, whereas (1384)^(1/2) in the exponential form. √1384 ≈ 37.201, 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 1384 is broken down into its prime factors.

Step 1: Finding the prime factors of 1384

Breaking it down, we get 2 x 2 x 2 x 173: 2^3 x 173^1

Step 2: Now we found out the prime factors of 1384. The second step is to make pairs of those prime factors. Since 1384 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 1384 using prime factorization is not straightforward.

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 1384, we need to group it as 84 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 9, which is less 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 84, which is the new dividend. Add the old divisor with the same number 3 + 3 to get 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and the 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 ≤ 484. Let us consider n as 7, now 6 x 7 x 7 = 294.

Step 6: Subtract 484 from 294; the difference is 190, 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 19000.

Step 8: Now we need to find the new divisor that is 744 because 744 x 4 = 2976.

Step 9: Subtracting 2976 from 19000, we get the result 16024.

Step 10: Now the quotient is 37.2

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

So the square root of √1384 is approximately 37.20.

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 1384 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1384. The smallest perfect square less than 1384 is 1369, and the largest perfect square greater than 1384 is 1444. √1384 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 (1384 - 1369) / (1444 - 1369) = 15 / 75 = 0.20 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 + 0.20 = 37.20.

So the square root of 1384 is approximately 37.20.

• 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 always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.

• Long Division Method: A method used to calculate the square root of a non-perfect square by dividing the number into groups of digits and finding the quotient step by step.

• Approximation Method: A method used to estimate the square root of a non-perfect square by comparing it with nearby perfect squares.