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 711.
The square root is the inverse of the square of the number. 711 is not a perfect square. The square root of 711 is expressed in both radical and exponential form. In the radical form, it is expressed as √711, whereas \(711^{1/2}\) is in the exponential form. √711 ≈ 26.678, 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 711 is broken down into its prime factors.
Step 1: Finding the prime factors of 711 Breaking it down, we get 3 x 237, and 237 can further be broken down into 3 x 79. Thus, the prime factorization is 3² x 79.
Step 2: Now we have found the prime factors of 711. The second step is to make pairs of those prime factors. Since 711 is not a perfect square, the digits of the number can’t be grouped in pairs to form a perfect square. Therefore, calculating √711 using prime factorization involves estimating between these factors.
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 711, we need to group it as 11 and 7.
Step 2: Now, we need to find a number n whose square is less than or equal to 7. We can say n is ‘2’ because 2 x 2 = 4, which is less than 7. The quotient is 2, and after subtracting 4 from 7, the remainder is 3.
Step 3: Bring down 11 to make the new dividend 311. Double the quotient and write it as 4.
Step 4: Find a digit x such that 4x x x is less than or equal to 311. Use trial and error to find x.
Step 5: Continue the steps to obtain a quotient with the desired decimal places. The result will be approximately 26.678.
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 711 using the approximation method.
Step 1: Now we have to find the closest perfect squares of √711. The smallest perfect square less than 711 is 676 (26²), and the largest perfect square greater than 711 is 729 (27²). √711 falls somewhere between 26 and 27.
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, (711 - 676) / (729 - 676) = 35 / 53 ≈ 0.6604. Adding this to the lower bound gives us the approximation: 26 + 0.6604 ≈ 26.6604. Thus, the approximate square root of 711 is 26.678.
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 √711?
The area of the square box is approximately 505.89 square units.
The area of the square = side².
The side length is given as √711.
Area of the square = side² = √711 x √711 = 26.678 x 26.678 ≈ 505.89.
Therefore, the area of the square box is approximately 505.89 square units.
A square-shaped building measuring 711 square feet is built; if each of the sides is √711, what will be the square feet of half of the building?
355.5 square feet
We can just divide the given area by 2 as the building is square-shaped.
Dividing 711 by 2, we get 355.5.
So half of the building measures 355.5 square feet.
Calculate √711 x 5.
Approximately 133.39
The first step is to find the square root of 711, which is approximately 26.678.
The second step is to multiply 26.678 by 5.
So 26.678 x 5 ≈ 133.39
What will be the square root of (711 + 9)?
The square root is 27.
To find the square root, we need to find the sum of (711 + 9). 711 + 9 = 720, and then √720 ≈ 27.
Therefore, the square root of (711 + 9) is approximately ±27.
Find the perimeter of the rectangle if its length ‘l’ is √711 units and the width ‘w’ is 38 units.
We find the perimeter of the rectangle as approximately 129.36 units.
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√711 + 38) = 2 × (26.678 + 38) ≈ 2 × 64.678 = 129.36 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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