Square Root of 623

The square root is the inverse of the square of the number. 623 is not a perfect square. The square root of 623 is expressed in both radical and exponential form. In the radical form, it is expressed as √623, whereas (623)^(1/2) in the exponential form. √623 ≈ 24.9583, 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. Now let us look at how 623 is broken down into its prime factors.

Step 1: Finding the prime factors of 623 Breaking it down, we get 7 x 89: 7^1 x 89^1

Step 2: Now we found out the prime factors of 623. The second step is to make pairs of those prime factors. Since 623 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 623 using prime factorization is not feasible for finding the exact square root.

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: To begin with, we need to group the numbers from right to left. In the case of 623, we need to consider it as 623.

Step 2: Find a number n whose square is less than or equal to 6. We can say n is 2 because 2 × 2 = 4 is less than 6. Now the quotient is 2 after subtracting 4 from 6, the remainder is 2.

Step 3: Bring down the next pair 23 to make the new dividend 223.

Step 4: Double the quotient obtained in Step 2, which is 2, to get 4, and place it as the new divisor's tens digit.

Step 5: Find a digit d such that 4d × d is less than or equal to 223. Upon trial, d is found to be 4 because 44 × 4 = 176.

Step 6: Subtract 176 from 223, resulting in a remainder of 47.

Step 7: Since the dividend is less than the divisor, add a decimal point and bring down two zeroes to make it 4700.

Step 8: Double the current quotient (24) to get 48, which will be part of the new divisor.

Step 9: Find a digit e such that 48e × e is less than or equal to 4700. Here, e is found to be 9 because 489 × 9 = 4401.

Step 10: Subtract 4401 from 4700, resulting in a remainder of 299.

Step 11: The quotient is now 24.9, and you can continue the process to get more decimal places. So the square root of √623 is approximately 24.958.

The approximation method is another method for finding 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 623 using the approximation method.

Step 1: Find the closest perfect squares around 623. The smallest perfect square less than 623 is 576 (24²), and the largest perfect square greater than 623 is 625 (25²). Thus, √623 falls somewhere between 24 and 25.

Step 2: Now apply the formula: (Given number - smallest perfect square) / (Largest perfect square - smallest perfect square) Using the formula: (623 - 576) / (625 - 576) = 47/49 ≈ 0.959 Adding this decimal to the lower bound: 24 + 0.959 = 24.959 Therefore, the approximate square root of 623 is 24.959.

• Square root: A square root is a value that, when multiplied by itself, gives the original number.

• Example: 4² = 16, and the inverse of the square is the square root, that 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.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 1, 4, 9, 16, and 25 are perfect squares.

• Prime factorization: The process of expressing a number as the product of its prime factors.

• Long division method: A method used to find the square root of a number by dividing the number into pairs of digits and performing division iteratively.