Last updated on May 26th, 2025
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in various fields like vehicle design, finance, etc. Here, we will discuss the 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:
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.
Students do make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in the long division method. Let us look at a few common mistakes in detail.
Can you help Max find the area of a square box if its side length is given as √623?
The area of the square is approximately 623 square units.
The area of the square = side².
The side length is given as √623.
Area of the square = side² = √623 × √623 = 623
Therefore, the area of the square box is approximately 623 square units.
A square-shaped building measuring 623 square feet is built. If each of the sides is √623, what will be the square feet of half of the building?
311.5 square feet
We can just divide the given area by 2 as the building is square-shaped. Dividing 623 by 2, we get 311.5.
So half of the building measures 311.5 square feet.
Calculate √623 × 5.
Approximately 124.79
First, find the square root of 623, which is approximately 24.959.
Then multiply 24.959 by 5. So, 24.959 × 5 ≈ 124.79.
What will be the square root of (600 + 23)?
The square root is approximately 24.959.
To find the square root, we calculate the sum of (600 + 23) = 623. The square root of 623 is approximately 24.959.
Find the perimeter of the rectangle if its length ‘l’ is √623 units and the width ‘w’ is 38 units.
We find the perimeter of the rectangle as approximately 125.916 units.
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√623 + 38) Perimeter = 2 × (24.959 + 38) ≈ 2 × 62.959 ≈ 125.916 units.
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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