Square Root of 14.5

The square root is the inverse of the square of the number. 14.5 is not a perfect square. The square root of 14.5 is expressed in both radical and exponential form. In the radical form, it is expressed as √14.5, whereas (14.5)^(1/2) in the exponential form. √14.5 ≈ 3.8079, 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 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. However, since 14.5 is not a perfect square and is a decimal, the prime factorization method cannot be directly applied as it is with whole numbers.

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: Group the numbers from right to left. For 14.5, it's already a single number.

Step 2: Identify a number n whose square is closest to 1. The closest perfect square is 1 (1×1), so the first digit of the quotient is 1.

Step 3: Subtract to get the remainder and bring down the decimal part to work with 450.

Step 4: Double the current quotient (1) to get 2, which will be used as the new divisor's first digit.

Step 5: Find a digit x such that 2x × x is less than or equal to 450. For x = 3, 23 × 3 = 69.

Step 6: Subtract 69 from 450 to get 381, and continue the process to get the decimal places.

Continue the steps until you achieve the desired precision, leading to an approximate square root of 3.8079.

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 14.5 using the approximation method.

Step 1: Identify the closest perfect squares to √14.5. The closest perfect squares are 9 (3×3) and 16 (4×4), so √14.5 is between 3 and 4.

Step 2: Apply the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square).

Using the formula: (14.5 - 9) / (16 - 9) = 5.5 / 7 ≈ 0.7857

Adding this to the lower integer: 3 + 0.7857 ≈ 3.7857

So the square root of 14.5 is approximately 3.8079.

• Square root: A square root is the inverse of a square. Example: 4² = 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.
   
• Decimal: A decimal is a number that includes a fractional component, such as 7.86, 8.65, and 9.42.
   
• Approximation: The process of finding a value that is close enough to the right answer, usually with some thought or calculation involved.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing the number into pairs and finding the most accurate quotient possible.