Square Root of 71

The square root is the inverse of the square of the number. 71 is not a perfect square. The square root of 71 is expressed in both radical and exponential form.

In the radical form, it is expressed as √71, whereas (71)^(1/2) is the exponential form. √71 ≈ 8.42615, 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, for non-perfect square numbers, the long-division method and approximation method are used. Let us now learn the following methods:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 71 is broken down into its prime factors. 71 is a prime number, so it cannot be broken down into smaller prime factors.

Therefore, calculating √71 using prime factorization is not feasible.

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 71, we need to group it as 71.

Step 2: Now we need to find n whose square is closest to 71 but less than or equal to 71. We can say n is ‘8’ because 8 × 8 = 64, which is less than 71. Now the quotient is 8, and after subtracting 64 from 71, the remainder is 7.

Step 3: Since the remainder is not zero, we need to add a decimal point to the quotient and bring down two zeroes to the remainder, making it 700.

Step 4: The new divisor will be the sum of the previous divisor multiplied by 2, i.e., 16, plus a new digit x such that 16x × x is less than or equal to 700. We find that x = 4 works because 164 × 4 = 656.

Step 5: Subtract 656 from 700, leaving a remainder of 44.

Step 6: Repeat the process by adding two more zeroes to make it 4400 and find a new divisor. Continue this process to get more decimal places. The square root of 71 is approximately 8.426.

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 71 using the approximation method.

Step 1: Now we have to find the closest perfect square of √71. The smallest perfect square less than 71 is 64, and the largest perfect square greater than 71 is 81. √71 falls somewhere between 8 and 9.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Applying the formula: (71 - 64) / (81 - 64) = 7 / 17 ≈ 0.412. 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: 8 + 0.412 ≈ 8.426, so the square root of 71 is approximately 8.426.

• 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.

• Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. Example: 71 is a prime number.

• Long division method: A method used to find more accurate values of square roots of numbers that are not perfect squares by performing long division.

• Approximation method: A method used to estimate the square root of a number by finding the nearest perfect squares and calculating the difference.