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:

• 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 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.

• Square root: A square root is the inverse of a square. 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.

• Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.

• Prime factorization: Prime factorization involves expressing a number as a product of its prime factors. For example, the prime factorization of 711 is 3² x 79.

• Long division method: The long division method is a technique for finding the square root of non-perfect squares by dividing the number into groups and using successive approximations.