Square Root of 21.25

The square root is the inverse operation of squaring a number. 21.25 is not a perfect square. The square root of 21.25 is expressed in both radical and exponential form. In radical form, it is expressed as √21.25, whereas in exponential form it is expressed as (21.25)^(1/2). √21.25 ≈ 4.609772, which is an irrational number because it cannot be expressed as a fraction of two integers.

The prime factorization method is typically used for perfect square numbers. However, for non-perfect squares like 21.25, we use methods such as the long division method and approximation method. Let us now learn the following methods:

• Long division method
• Approximation method

The long division method is particularly useful for finding the square root of non-perfect square numbers. Let us learn how to find the square root of 21.25 using this method, step by step:

Step 1: Pair digits from right to left, starting with the decimal point. For 21.25, group it as 21 and 25.

Step 2: Find the largest number whose square is less than or equal to 21. This number is 4 because 4 x 4 = 16, which is less than 21.

Step 3: Subtract 16 from 21, giving a remainder of 5. Bring down 25 to make it 525.

Step 4: Double the quotient (4) to get 8, which becomes the starting digit of the new divisor.

Step 5: Find a digit X such that 8X multiplied by X gives a product less than or equal to 525. Here, 86 x 6 = 516.

Step 6: Subtract 516 from 525 to get 9. Add decimal points and bring down 00 to make it 900.

Step 7: Continue the process with the new divisor, 92, to refine the square root to desired decimal places.

The square root of 21.25 is approximately 4.609772.

The approximation method provides an easy way to estimate the square root of a given number. Here’s how to use it for 21.25:

Step 1: Identify the perfect squares closest to 21.25.

These are 16 (4^2) and 25 (5^2).

Therefore, √21.25 is between 4 and 5.

Step 2: Apply the formula:

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

For 21.25, use (21.25 - 16) / (25 - 16) = 5.25 / 9 ≈ 0.5833.

Step 3: Add this value to the lower bound of the range: 4 + 0.5833 ≈ 4.5833.

So, the approximate square root of 21.25 is 4.609772.

• Square root: The square root of a number is a value that, when multiplied by itself, gives the original number. For example, √16 = 4.
   
• Irrational number: A number that cannot be written as a simple fraction; its decimal goes on forever without repeating. Example: √21.25 is irrational.
   
• Approximation method: A technique for estimating the value of a square root by comparing it to nearby perfect squares.
   
• Long division method: A systematic approach to find the exact square root of a non-perfect square number through division.
   
• Decimal: A number that includes a whole number and a fraction expressed with a dot. For example, 4.609772 is a decimal.