Square Root of 194.75

The square root is the inverse of the square of the number. 194.75 is not a perfect square. The square root of 194.75 is expressed in both radical and exponential form. In the radical form, it is expressed as √194.75, whereas (194.75)^(1/2) in the exponential form. √194.75 ≈ 13.954, 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. Since 194.75 is not an integer, it does not have a straightforward prime factorization like whole numbers. Therefore, calculating 194.75 using prime factorization is not applicable.

The long division method is particularly used for non-perfect square numbers. 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 pairs. In the case of 194.75, consider it as 19475 (ignoring the decimal for now).

Step 2: Determine the largest number whose square is less than or equal to 1. Here, it is 1.

Step 3: Subtract 1 from 1 to get a remainder of 0 and bring down 94 to make it 94.

Step 4: Double the divisor (which is 1) to get 2, and find a number n such that 2n × n ≤ 94. Choose n = 4, as 24 × 4 = 96, which is greater than 94. So, n = 3, giving us 23 × 3 = 69.

Step 5: Subtract 69 from 94 to get a remainder of 25. Bring down 75 to make it 2575.

Step 6: Double the current quotient (13) to get 26, and find n such that 26n × n ≤ 2575. Choose n = 9, as 269 × 9 = 2421.

Step 7: Subtract 2421 from 2575 to get 154.

Step 8: Continue this process, adding zeros in pairs to the remainder and repeating steps to obtain more decimal places.

The square root of 194.75 is approximately 13.954.

The approximation method is another method for finding square roots, and 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 194.75 using the approximation method.

Step 1: Identify the closest perfect squares around 194.75. The smallest perfect square less than 194.75 is 169, and the largest perfect square greater than 194.75 is 225.

Step 2: Since 194.75 falls between 169 (13^2) and 225 (15^2), it is between 13 and 15. More precisely, between 13.5 and 14.

Step 3: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). (194.75 - 169) / (225 - 169) ≈ 0.456

Step 4: Add this decimal to the lower bound (13) to approximate the square root. 13 + 0.954 ≈ 13.954

Thus, the approximate square root of 194.75 is 13.954.

• 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, which 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, the positive square root is more commonly used due to its applications in the real world. This is known as 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.

• Perfect square: A perfect square is a number that can be expressed as the square of an integer, for example: 1, 4, 9, 16, 25, etc.