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 the field of vehicle design, finance, etc. Here, we will discuss the square root of 457.
The square root is the inverse of the square of the number. 457 is not a perfect square. The square root of 457 is expressed in both radical and exponential form. In the radical form, it is expressed as √457, whereas (457)^(1/2) in the exponential form. √457 ≈ 21.385, 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 457 is broken down into its prime factors.
Step 1: Finding the prime factors of 457. Breaking it down, we get 457 is a prime number itself.
Step 2: Since 457 is a prime number, it cannot be factored further.
Therefore, calculating 457 using prime factorization is not applicable.
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 457, we need to group it as 57 and 4.
Step 2: Now we need to find n whose square is 4. We can say n as ‘2’ because 2 × 2 is equal to 4. Now the quotient is 2, and after subtracting 4-4, the remainder is 0.
Step 3: Now let us bring down 57, which is the new dividend. Add the old divisor with the same number 2 + 2, we get 4, which will be our new divisor.
Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 4n as the new divisor, we need to find the value of n.
Step 5: The next step is finding 4n × n ≤ 57. Let us consider n as 1, now 41 × 1 = 41.
Step 6: Subtract 57 from 41, the difference is 16, and the quotient is 21.
Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 1600.
Step 8: Now we need to find the new divisor that is 425 because 425 × 3 = 1275.
Step 9: Subtracting 1275 from 1600, we get the result 325.
Step 10: Now the quotient is 21.3.
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.
So the square root of √457 is approximately 21.39.
The approximation method is another method for finding the 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 457 using the approximation method.
Step 1: Now we have to find the closest perfect square of √457. The smallest perfect square less than 457 is 441, and the largest perfect square greater than 457 is 484. √457 falls somewhere between 21 and 22.
Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (457 - 441) ÷ (484-441) = 16/43 ≈ 0.372.
Using the formula, we identified the decimal point of our square root.
The next step is adding the value we got initially to the decimal number, which is 21 + 0.372 = 21.372, so the square root of 457 is approximately 21.372.
Students do make mistakes while finding the square root, like forgetting about the negative square root, skipping long division methods, etc. Now let us look at a few of those mistakes that students tend to make in detail.
Can you help Max find the area of a square box if its side length is given as √457?
The area of the square is approximately 457 square units.
The area of the square = side².
The side length is given as √457.
Area of the square = side² = √457 × √457 = 457.
Therefore, the area of the square box is approximately 457 square units.
A square-shaped building measuring 457 square feet is built; if each of the sides is √457, what will be the square feet of half of the building?
228.5 square feet
We can just divide the given area by 2 as the building is square-shaped.
Dividing 457 by 2 = we get 228.5.
So half of the building measures 228.5 square feet.
Calculate √457 × 5.
Approximately 106.925
The first step is to find the square root of 457, which is approximately 21.385, the second step is to multiply 21.385 with 5. So 21.385 × 5 ≈ 106.925.
What will be the square root of (457 - 17)?
The square root is approximately 21.
To find the square root, we need to find the difference of (457 - 17). 457 - 17 = 440, and then the square root of 440 is approximately 20.976.
Therefore, the square root of (457 - 17) is approximately 21.
Find the perimeter of the rectangle if its length ‘l’ is √457 units and the width ‘w’ is 40 units.
We find the perimeter of the rectangle as approximately 122.77 units.
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√457 + 40) ≈ 2 × (21.385 + 40) ≈ 2 × 61.385 ≈ 122.77 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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