Square Root of 1215

The square root is the inverse of the square of the number. 1215 is not a perfect square. The square root of 1215 is expressed in both radical and exponential form. In the radical form, it is expressed as √1215, whereas (1215)^(1/2) in the exponential form. √1215 ≈ 34.848, 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 1215 is broken down into its prime factors.

Step 1: Finding the prime factors of 1215 Breaking it down, we get 3 × 3 × 3 × 3 × 5 × 3: 3^5 × 5

Step 2: Now we found out the prime factors of 1215. The second step is to make pairs of those prime factors. Since 1215 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1215 using prime factorization is impossible.

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 1215, we need to group it as 15 and 12.

Step 2: Now we need to find n whose square is less than or equal to 12. We can say n as '3' because 3 × 3 = 9 is less than 12. Now the quotient is 3 after subtracting 9 from 12, the remainder is 3.

Step 3: Now let us bring down 15, which is the new dividend. Add the old divisor with the same number 3 + 3 we get 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 ≤ 315. Let us consider n as 5, now 65 × 5 = 325, which is greater than 315. Try n as 4, 64 × 4 = 256.

Step 6: Subtract 256 from 315, the difference is 59, and the quotient is 34.

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 5900.

Step 8: Now we need to find the new divisor that is 698 because 698 × 8 = 5584.

Step 9: Subtracting 5584 from 5900, we get the result 316.

Step 10: Now the quotient is 34.8.

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

So the square root of √1215 is approximately 34.85.

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 1215 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1215. The smallest perfect square less than 1215 is 1156, and the largest perfect square greater than 1215 is 1225. √1215 falls somewhere between 34 and 35.

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 (1215 - 1156) / (1225 - 1156) = 59 / 69 ≈ 0.855. 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 which is 34 + 0.855 ≈ 34.855, so the square root of 1215 is approximately 34.855.

• 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, 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: The process of expressing a number as a product of its prime numbers. For example, the prime factorization of 18 is 2 × 3 × 3.
   
• Long division method: A technique used to find the square root of non-perfect squares by dividing the number step by step.