Square Root of 650.25

The square root is the inverse of the square of the number. 650.25 is not a perfect square. The square root of 650.25 is expressed in both radical and exponential form. In the radical form, it is expressed as √650.25, whereas (650.25)^(1/2) in the exponential form. √650.25 = 25.5, which is a rational number because it can 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 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. 650.25 can be expressed as a product of its prime factors:

Step 1: Express 650.25 as a fraction: 650.25 = 65025/100

Step 2: Find the prime factors of the numerator, 65025: 65025 = 5 × 5 × 5 × 5 × 13 × 13

Step 3: Find the prime factors of the denominator, 100: 100 = 2 × 2 × 5 × 5

Step 4: Simplify the expression: √(65025/100) = (5 × 5 × 13)/(2 × 5) = 25.5

As a result, the square root of 650.25 is 25.5, which is a rational number.

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: Pair the digits from right to left. For 650.25, this results in the pairs 06|50|25.

Step 2: Find the largest number whose square is less than or equal to 6. The number is 2. Subtract 4 from 6, and bring down the next pair (50), giving 250.

Step 3: Double the divisor (2) giving 4, and find a digit X such that 4X × X is less than or equal to 250. The number is 6. Subtract 246 from 250, giving 4.

Step 4: Bring down the next pair (25), giving 425. Double the divisor (26) giving 52, and find a digit X such that 52X × X is less than or equal to 425. The number is 8. Subtract 424 from 425, giving 1.

Step 5: Since we have considered only up to two decimal places, the square root of 650.25 is approximately 25.5.

The approximation method is another way to find 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 650.25 using the approximation method.

Step 1: Identify the closest perfect squares around 650.25. The closest perfect square less than 650.25 is 625, and the closest perfect square greater than 650.25 is 676.

Step 2: √650.25 falls between √625 (25) and √676 (26).

Step 3: Use interpolation to approximate: (650.25 - 625)/(676 - 625) = 0.5

Step 4: Add this to the lower bound: 25 + 0.5 = 25.5

Therefore, the square root of 650.25 is approximately 25.5.

• Square root: The square root is the inverse of a square. For example, 5^2 = 25, and the inverse of the square is the square root, that is √25 = 5.
   
• Rational number: A rational number is a number that can be expressed as a fraction p/q, where q is not equal to zero, and p and q are integers.
   
• Irrational number: An irrational number cannot be written as a simple fraction because it is a non-repeating, non-terminating decimal. Examples include √2 and π.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example, 16 is a perfect square because 4 × 4 = 16.
   
• Long division method: A method used to find the square root of a number by dividing it into pairs of digits from right to left and finding the closest perfect square.