Square Root of 381

The square root is the inverse of the square of a number. 381 is not a perfect square. The square root of 381 is expressed in both radical and exponential form. In the radical form, it is expressed as √381, whereas (381)^(1/2) in exponential form. √381 ≈ 19.521, 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 and approximation methods 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 381 is broken down into its prime factors.

Step 1: Finding the prime factors of 381 Breaking it down, we get 3 x 127.

Step 2: Now we found out the prime factors of 381. The second step is to make pairs of those prime factors. Since 381 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √381 using prime factorization is not directly possible.

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 381, we need to group it as 81 and 3.

Step 2: Now we need to find n whose square is 3. We can say n as ‘1’ because 1 x 1 is less than or equal to 3. Now the quotient is 1, and after subtracting 1 from 3, the remainder is 2.

Step 3: Now let us bring down 81, which is the new dividend. Add the old divisor with the same number 1 + 1, we get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 281. Let us consider n as 9, now 29 x 9 = 261.

Step 6: Subtract 261 from 281; the difference is 20, and the quotient is 19.

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 2000.

Step 8: Now we need to find the new divisor that is 391, because 391 x 5 = 1955.

Step 9: Subtracting 1955 from 2000, we get the result 45.

Step 10: Now the quotient is 19.5.

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 √381 ≈ 19.52.

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 381 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √381. The smallest perfect square less than 381 is 361, and the largest perfect square more than 381 is 400. √381 falls somewhere between 19 and 20.

Step 2: Now we need to apply the formula that is

(Given number - smallest perfect square) / (Larger perfect square - smallest perfect square).

Going by the formula (381 - 361) / (400 - 361) = 0.51.

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 19 + 0.51 = 19.51, so the square root of 381 is approximately 19.51.

• 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.
   
• Prime factorization: The process of breaking down a number into its basic prime factors.
   
• Long division method: A method used to find the square root of non-perfect squares through division.
   
• Approximation method: A technique used to estimate the square root of a number by finding the closest perfect squares.