Square Root of 1015

A square root is the inverse of the square of a number. 1015 is not a perfect square. The square root of 1015 is expressed in both radical and exponential forms. In the radical form, it is expressed as √1015, whereas (1015)^(1/2) is its exponential form. √1015 ≈ 31.8591, 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 employed. Let us now learn these methods:

• Prime factorization method
• Long division method 
• Approximation method

The product of prime factors is the prime factorization of a number. Now let us examine how 1015 is broken down into its prime factors.

Step 1: Finding the prime factors of 1015 Breaking it down, we get 5 x 7 x 29.

Step 2: Now we found out the prime factors of 1015. The second step is to make pairs of those prime factors. Since 1015 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating 1015 using prime factorization to find an exact square root is not possible.

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 1015, we need to group it as 15 and 10.

Step 2: Now we need to find n whose square is closest to 10. We can say n as ‘3’ because 3 x 3 = 9 is lesser than or equal to 10. Now the quotient is 3, after subtracting 9 from 10 the remainder is 1.

Step 3: Now let us bring down 15 to make the new dividend 115. Add the old divisor with the same number, 3 + 3 = 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 115. Let us consider n as 1, now 61 x 1 = 61.

Step 6: Subtract 61 from 115, the difference is 54, and the quotient is 31.

Step 7: 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 5400.

Step 8: Now we need to find the new divisor that is 638 because 638 x 8 = 5104.

Step 9: Subtracting 5104 from 5400, we get the result 296.

Step 10: Now the quotient is 31.8.

Step 11: Continue doing these steps until we get two numbers after the decimal point. If there are no more decimal values, continue till the remainder is zero.

So the square root of √1015 ≈ 31.86.

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 1015 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √1015. The smallest perfect square less than 1015 is 961 (which is 31^2), and the largest perfect square more than 1015 is 1024 (which is 32^2). √1015 falls somewhere between 31 and 32.

Step 2: Now we need to apply the formula that is: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (1015 - 961) / (1024 - 961) = 54 / 63 ≈ 0.857. Using the formula, we identified the decimal point to add to our initial approximation. The next step is adding the value we got initially to the decimal number: 31 + 0.857 ≈ 31.857.

So the square root of 1015 is approximately 31.86.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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 the principal square root.
   
• Factors: Factors of a number are whole numbers that divide it completely without leaving a remainder.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example, 16 is a perfect square because it is 4^2.