Square Root of 1394

The square root is the inverse of the square of the number. 1394 is not a perfect square. The square root of 1394 is expressed in both radical and exponential form. In the radical form, it is expressed as √1394, whereas (1394)^(1/2) in the exponential form. √1394 ≈ 37.313, 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 1394 is broken down into its prime factors.

Step 1: Finding the prime factors of 1394

Breaking it down, we get 2 x 19 x 73: 2^1 x 19^1 x 73^1

Step 2: Now we found out the prime factors of 1394. The second step is to make pairs of those prime factors. Since 1394 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 1394 using prime factorization does not yield a perfect square root.

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 1394, we need to group it as 94 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 94, which is the new dividend. 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 current quotient with the number we add in the next step. Now we look for a digit n such that 6n x n is less than or equal to 494.

Step 5: The next step is finding 6n x n ≤ 494. Let us consider n as 8, now 68 x 8 = 544, which is too high, so we try n as 7, then 67 x 7 = 469.

Step 6: Subtracting 469 from 494, the difference is 25, 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 2500.

Step 8: Now we need to find the new divisor that is 746 because 746 x 3 = 2238.

Step 9: Subtracting 2238 from 2500, we get the result 262.

Step 10: Now the quotient is 37.3

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 √1394 ≈ 37.313

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 1394 using the approximation method.

Step 1: Now we find the closest perfect squares around √1394. The smallest perfect square less than 1394 is 1369, and the largest perfect square greater than 1394 is 1444. √1394 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 (1394 - 1369) / (1444 - 1369) ≈ 0.333 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 + 0.333 ≈ 37.333.

So the square root of 1394 is approximately 37.313.

• 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 a principal square root.

• Prime factorization: Breaking down a number into its basic prime number factors, which are the building blocks of the number.

• Long division method: A step-by-step process for dividing complex numbers to find square roots, especially useful for non-perfect squares. ```