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 fields such as vehicle design, finance, etc. Here, we will discuss the square root of 2057.
The square root is the inverse of the square of the number. 2057 is not a perfect square. The square root of 2057 is expressed in both radical and exponential form. In the radical form, it is expressed as √2057, whereas (2057)^(1/2) in exponential form. √2057 ≈ 45.342, 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 2057 is broken down into its prime factors.
Step 1: Finding the prime factors of 2057
Breaking it down, we get 2057 = 11 x 11 x 17.
Step 2: Now we have found the prime factors of 2057. The second step is to make pairs of those prime factors. Since 2057 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 2057 using prime factorization is impossible.
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 2057, we group it as 57 and 20.
Step 2: Now we need to find n whose square is ≤ 20. We can say n as ‘4’ because 4 x 4 = 16 and is lesser than or equal to 20. Now the quotient is 4; after subtracting 16 from 20, the remainder is 4.
Step 3: Now let us bring down 57, which is the new dividend. Add the old divisor, 4, with the same number 4 + 4, we get 8, which will be our new divisor.
Step 4: The new divisor is 8, and we need to find n such that 8n x n ≤ 457. Let us consider n as 5, now 85 x 5 = 425.
Step 5: Subtract 425 from 457, the difference is 32, and the quotient is 45.
Step 6: 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 3200.
Step 7: Now we need to find the new divisor that is 453 because 453 x 7 = 3171.
Step 8: Subtracting 3171 from 3200, we get the result 29.
Step 9: 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 √2057 is approximately 45.34.
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 2057 using the approximation method.
Step 1: Now we have to find the closest perfect squares of √2057. The smallest perfect square less than 2057 is 2025, and the largest perfect square greater than 2057 is 2116. √2057 falls somewhere between 45 and 46.
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 (2057 - 2025) / (2116 - 2025) = 32 / 91 ≈ 0.35.
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 45 + 0.35 = 45.35. So the square root of 2057 is approximately 45.35.
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 √2057?
The area of the square is 2057 square units.
The area of the square = side².
The side length is given as √2057.
Area of the square = side² = √2057 x √2057 = 2057.
Therefore, the area of the square box is 2057 square units.
A square-shaped building measuring 2057 square feet is built; if each of the sides is √2057, what will be the square feet of half of the building?
1028.5 square feet
We can just divide the given area by 2 as the building is square-shaped.
Dividing 2057 by 2 = we get 1028.5.
So half of the building measures 1028.5 square feet.
Calculate √2057 x 3.
136.03
The first step is to find the square root of 2057, which is approximately 45.34.
The second step is to multiply 45.34 by 3.
So 45.34 x 3 ≈ 136.03.
What will be the square root of (2025 + 32)?
The square root is approximately 45.35.
To find the square root, we need to find the sum of (2025 + 32). 2025 + 32 = 2057, and then √2057 ≈ 45.35.
Therefore, the square root of (2025 + 32) is approximately ±45.35.
Find the perimeter of the rectangle if its length ‘l’ is √2057 units and the width ‘w’ is 25 units.
We find the perimeter of the rectangle as 140.68 units.
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√2057 + 25) = 2 × (45.34 + 25) = 2 × 70.34 = 140.68 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.
: He loves to play the quiz with kids through algebra to make kids love it.