Square Root of 197

The square root is the inverse of the square of the number. 197 is not a perfect square. The square root of 197 is expressed in both radical and exponential form. In the radical form, it is expressed as √197, whereas in exponential form it is expressed as (197)^(1/2). √197 ≈ 14.03567, 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, the prime factorization method is not used; instead, 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 197 is broken down into its prime factors.

Step 1: Finding the prime factors of 197 197 is a prime number itself, so it cannot be broken down further into other prime factors.

Step 2: Since 197 is not a perfect square and it is a prime number, calculating √197 using the prime factorization method is not feasible.

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 197, we need to group it as 97 and 1.

Step 2: Now we need to find n whose square is 1. We can say n as ‘1’ because 1 × 1 is lesser 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 97, 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, we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 97. Let us consider n as 4, now 24 × 4 = 96.

Step 6: Subtract 97 from 96; the difference is 1, 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 100.

Step 8: Now we need to find the new divisor, which is 280 because 280 × 0 = 0.

Step 9: Subtracting 0 from 100, we get the result 100.

Step 10: 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 √197 is approximately 14.04.

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 197 using the approximation method.

Step 1: Now we have to find the closest perfect square of √197. The smallest perfect square less than 197 is 196, and the largest perfect square greater than 197 is 225. √197 falls somewhere between 14 and 15.

Step 2: Now we need to apply the linear approximation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using this formula: (197 - 196) / (225 - 196) ≈ 0.03448. 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 14 + 0.03448 ≈ 14.03448.

So the square root of 197 is approximately 14.035.

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

• Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. It has only two factors: 1 and itself.

• Decimal: A decimal is a number that consists of a whole number and a fractional part separated by a decimal point. For example: 7.86, 8.65, and 9.42 are decimals.

• Exponential form: The exponential form is a way of expressing numbers that have been raised to a power. It is represented as a base raised to an exponent, for example, 2³ = 8.