Square Root of 565.44

The square root is the inverse of the square of the number. 565.44 is not a perfect square, but it is a rational number. The square root of 565.44 is expressed in both radical and exponential form. In the radical form, it is expressed as √565.44, whereas (565.44)^(1/2) in the exponential form. √565.44 = 23.78, 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, for numbers with decimals like 565.44, methods such as the long-division method and approximation method are used. Let us now learn the following methods:

1. Long division method
2. Approximation method

The long division method is particularly used for numbers that are not perfect squares. 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: To begin with, we need to group the digits of 565.44 into pairs from the decimal point. So, we have groups: 56 and 54, and 44 after the decimal.

Step 2: Find the largest number whose square is less than or equal to 56. This number is 7 because 7^2 = 49, which is less than 56. The quotient is 7, and the remainder is 56 - 49 = 7.

Step 3: Bring down the next pair, 54, making the new dividend 754.

Step 4: Double the quotient and use it as the new divisor. So, 2 × 7 = 14. We now need to find a digit x such that 14x × x is less than or equal to 754. The suitable digit is 5 because 145 × 5 = 725.

Step 5: Subtract 725 from 754, giving a remainder of 29. Bring down the next pair of digits, 44, to make it 2944.

Step 6: Repeat the process: double the new quotient (75), giving 150. Find a digit x such that 150x × x ≤ 2944. The digit is 9 because 1509 × 9 = 13581. However, we must adjust to ensure that the calculation aligns with the process.

Step 7: Calculate further to attain desired precision. After repeating these steps, the quotient converges to 23.78.

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 565.44 using the approximation method.

Step 1: Find the closest perfect squares of 565.44. The closest smaller perfect square is 529 (23²), and the closest larger perfect square is 576 (24²). √565.44 falls somewhere between 23 and 24

Step 2: Apply interpolation for more precision. Using the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square), we get: (565.44 - 529) ÷ (576 - 529) = 0.78.

Step 3: Add this decimal to the smaller perfect square root: 23 + 0.78 = 23.78.

Therefore, the square root of 565.44 is approximately 23.78.

• Square root: A square root is the value that, when multiplied by itself, gives the original number. Example: √16 = 4 because 4 × 4 = 16.

• Rational number: A rational number can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero.

• Decimal: A decimal is a number that consists of a whole number and a fractional part separated by a decimal point, such as 23.78.

• Long division method: A technique used to find the square root of non-perfect squares, involving iterative calculations and refinement.

• Perfect square: A number that is the square of an integer. Example: 25 is a perfect square because it is 5².