Square Root of 191

The square root is the inverse of the square of the number. 191 is not a perfect square. The square root of 191 is expressed in both radical and exponential form. In the radical form, it is expressed as √191, whereas (191)^(1/2) in the exponential form. √191 ≈ 13.82027, 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, for non-perfect square numbers like 191, the prime factorization method is not applicable. Instead, the long division method and approximation method are used. Let us now learn the following methods:

• Long division method
   
• Approximation method

The long division method is particularly used for non-perfect square numbers. In this method, we should find 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 191, we need to group it as 91 and 1.

Step 2: Now we need to find n whose square is less than or equal to 1. We can say n is '1' because 1 × 1 is less than or equal to 1. Now the quotient is 1; after subtracting 1 - 1, the remainder is 0.

Step 3: Now let us bring down 91, which is the new dividend. Add the old divisor with the same number: 1 + 1 = 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; we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 91. Let us consider n as 4, now 2 × 4 × 4 = 32.

Step 6: Subtract 91 from 32, the difference is 59, and the quotient is 14.

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 5900.

Step 8: Now we need to find the new divisor, which will be 289, because 289 × 9 = 2601.

Step 9: Subtracting 2601 from 5900, we get the result 3299.

Step 10: Now the quotient is 13.8.

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 √191 is approximately 13.82.

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 191 using the approximation method.

Step 1: Now we have to find the closest perfect square of √191. The smallest perfect square less than 191 is 169, and the largest perfect square greater than 191 is 196. √191 falls somewhere between 13 and 14.

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 (191 - 169) / (196 - 169) ≈ 0.815. 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 13 + 0.815 ≈ 13.82, so the square root of 191 is approximately 13.82.

• Square root: A square root is the inverse of a square. Example: 4² = 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 p/q, where q is not equal to zero, and both p and q are integers.

• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

• Decimal: A decimal is a number that contains a whole number and a fractional part separated by a decimal point. Examples include 7.86, 8.65, and 9.42.

• Long division method: A method used to find the square root of non-perfect square numbers by dividing the number into groups and finding the quotient step by step.