Square Root of 2017

The square root is the inverse of the square of a number. 2017 is not a perfect square. The square root of 2017 is expressed in both radical and exponential form. In the radical form, it is expressed as √2017, whereas in exponential form it is (2017)^(1/2). √2017 ≈ 44.911, 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 prime factorization method is not applicable, so we use the long-division method and approximation method. Let us now learn the following methods: 

• Long division method
• Approximation method

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 2017, group it as 20 and 17.

Step 2: Now, find n whose square is less than or equal to 20. We can say n as '4' because 4 × 4 = 16 is less than 20. The quotient is 4; after subtracting 16 from 20, the remainder is 4.

Step 3: Bring down 17 to make it 417. Add the old divisor (4) with the same number (4) to get 8, which will be our new divisor.

Step 4: The new divisor becomes 8n. We need to find the value of n such that 8n × n ≤ 417. Let us consider n as 5; now 85 × 5 = 425, which is more than 417. Try n as 4; 84 × 4 = 336.

Step 5: Subtract 336 from 417; the difference is 81. The quotient is 44.

Step 6: Since 81 is the remainder, we need to add a decimal point and continue the process. Add two zeroes to make it 8100.

Step 7: Now find the new divisor, which is 889, because 889 × 9 = 8001.

Step 8: Subtracting 8001 from 8100 gives us 99.

Step 9: Continue doing these steps until we get two numbers after the decimal point.

Thus, the square root of √2017 is approximately 44.91.

The approximation method is another method for finding square roots, and it is an easy method for estimating the square root of a given number. Let us learn how to find the square root of 2017 using the approximation method.

Step 1: Find the closest perfect squares around √2017. The smallest perfect square less than 2017 is 1936 (44^2), and the largest perfect square greater than 2017 is 2025 (45^2). √2017 lies between 44 and 45.

Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). (2017 - 1936) / (2025 - 1936) = 81 / 89 ≈ 0.91 Adding this decimal to the lower bound gives 44 + 0.91 = 44.91, so the square root of 2017 is approximately 44.91.

• Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4.
   
• Irrational number: An irrational number cannot be expressed as a fraction of two integers. Its decimal form is non-repeating and non-terminating.
   
• Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
   
• Decimal approximation: A decimal approximation is an estimate of a number using decimal places to represent its value accurately.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing the number and refining the result step by step.