Square Root of 199.68

The square root is the inverse operation of squaring a number. 199.68 is not a perfect square. The square root of 199.68 can be expressed in both radical and exponential forms. In the radical form, it is expressed as √199.68, whereas in the exponential form, it is expressed as (199.68)^(1/2). √199.68 ≈ 14.1337, which is an irrational number because it cannot be expressed as a simple fraction.

The prime factorization method is generally used for perfect square numbers. However, for non-perfect square numbers like 199.68, the long division method and approximation method are used. Let us learn the following methods:

• Prime factorization method
   
• Long division method
   
• Approximation method

The prime factorization of a number is the product of its prime factors. Since 199.68 is not a whole number, prime factorization is not applicable in the traditional sense. Therefore, calculating the square root of 199.68 using prime factorization is not feasible.

The long division method is particularly useful for non-perfect square numbers. Here is how you can find the square root using this method:

Step 1: Begin by grouping the numbers from right to left. For 199.68, group it as 68 and 199.

Step 2: Find a number whose square is less than or equal to 199. We can use 14 because 14 x 14 = 196, which is less than 199. The quotient is 14, and the remainder is 3.

Step 3: Bring down 68, making the new dividend 368. Double the quotient (14), giving us 28 as part of the new divisor.

Step 4: Find a number n such that 28n x n is less than or equal to 368. Using n = 1, we have 281 x 1 = 281.

Step 5: Subtract 281 from 368, leaving a remainder of 87. Step 6: Since the remainder is less than the new divisor, introduce a decimal point and add two zeros, making the new dividend 8700.

Step 7: Continue the process until the desired precision is achieved. The quotient will approximate √199.68 to two decimal places as 14.14.

The approximation method is another way to find square roots. Here's how to find the square root of 199.68 this way:

Step 1: Identify the closest perfect squares around 199.68. The smallest perfect square is 196, and the largest is 225. √199.68 falls between √196 (14) and √225 (15).

Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). Applying the formula: (199.68 - 196) / (225 - 196) = 3.68 / 29 ≈ 0.127.

Step 3: Add this to the smaller square root: 14 + 0.127 ≈ 14.127.

Therefore, √199.68 is approximately 14.13.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: √16 = 4.

• Irrational number: A number that cannot be expressed as a ratio of two integers. Example: √2.

• Long division method: A method used to find the square root of non-perfect squares by dividing the number into pairs from right to left.

• Approximation method: A technique to estimate the square root of a number by using nearby perfect squares.

• Decimal: A number that includes a fractional part separated from the integer part by a decimal point. Example: 3.14.