Square Root of 1378

The square root is the inverse of the square of the number. 1378 is not a perfect square. The square root of 1378 is expressed in both radical and exponential form. In radical form, it is expressed as √1378, whereas (1378)^(1/2) in the exponential form. √1378 ≈ 37.122, 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 the 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 1378 is broken down into its prime factors.

Step 1: Finding the prime factors of 1378 1378 can be broken down as 2 x 689. Further breaking down 689, we get 689 = 13 x 53. Therefore, 1378 = 2 x 13 x 53.

Step 2: Now we found out the prime factors of 1378. The second step is to make pairs of those prime factors. Since 1378 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating √1378 using prime factorization does not yield an exact value.

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 1378, we need to group it as 78 and 13.

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

Step 3: Now let us bring down 78, 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 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 ≤ 478. Let us consider n as 7. Now 67 x 7 = 469.

Step 6: Subtract 469 from 478; the difference is 9, 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 900.

Step 8: Now we need to find the new divisor, which is 371 because 371 x 2 = 742.

Step 9: Subtracting 742 from 900, we get the result 158.

Step 10: Now the quotient is 37.12

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

So the square root of √1378 is approximately 37.12.

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 1378 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1378. The smallest perfect square less than 1378 is 1369 (37^2), and the largest perfect square greater than 1378 is 1444 (38^2). √1378 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). Using the formula (1378 - 1369) ÷ (1444 - 1369) ≈ 0.12. 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 + 0.12 = 37.12, so the square root of 1378 is approximately 37.12.

• 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 is most commonly used due to its applications in the real world. That is why it is called the principal square root.

• 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.

• Prime factorization: The process of dividing a number into its basic prime factors. For example, the prime factorization of 1378 is 2 x 13 x 53.