Square Root of 25.25

The square root is the inverse of the square of the number. 25.25 is not a perfect square. The square root of 25.25 is expressed in both radical and exponential form. In the radical form, it is expressed as √25.25, whereas (25.25)^(1/2) in the exponential form. √25.25 = 5.0249378106, 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 long-division method and approximation method are used. Let us now learn the following methods:

• Long division method
• Approximation method

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: Start by grouping the digits from right to left in pairs. For 25.25, consider 25 and 25 as separate groups.

Step 2: Find a number whose square is less than or equal to 25. The number is 5, since 5 × 5 = 25. The quotient is 5, and the remainder is 0.

Step 3: Bring down the next pair, which is 25, and place it next to the remainder, making it 025. Add the previous divisor, 5, to itself to get 10.

Step 4: Find a number (n) such that 10n × n ≤ 25. The number is 2, as 102 × 2 = 204, which is less than 250.

Step 5: Subtract 204 from 250, leaving a remainder of 46.

Step 6: Bring down another pair of zeros to make the number 4600.

Step 7: Continue the process until you reach the desired precision.

The quotient, up to two decimal places, is 5.02.

The approximation method is another way to find 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 25.25 using the approximation method.

Step 1: Identify the closest perfect squares to 25.25. The closest perfect squares are 25 (5²) and 36 (6²).

Step 2: Apply the formula:

(Given number - smaller perfect square) ÷ (larger perfect square - smaller perfect square).

Step 3: Using the formula, (25.25 - 25) ÷ (36 - 25) = 0.25 ÷ 11 = 0.0227.

Step 4: Add this to the square root of the smaller perfect square: 5 + 0.0227 = 5.0227.

This approximation gives the square root of 25.25 as approximately 5.024.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. Example: √25.25 ≈ 5.0249.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction. It has a non-repeating, non-terminating decimal expansion.
   
• Principal square root: The principal square root is the positive square root of a number. For example, the principal square root of 25.25 is 5.0249378106.
   
• Decimal: A decimal is a fraction written in a special form. For example, 0.5 is a decimal and is equal to 1/2.
   
• Approximation: An approximation is a value or quantity that is nearly but not exactly correct. It is used to estimate values when precise figures are not necessary.