Square Root of 1245

The square root is the inverse of the square of the number. 1245 is not a perfect square. The square root of 1245 is expressed in both radical and exponential form. In the radical form, it is expressed as √1245, whereas (1245)^(1/2) in the exponential form. √1245 ≈ 35.268, 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 1245 is broken down into its prime factors.

Step 1: Finding the prime factors of 1245 Breaking it down, we get 3 x 5 x 83: 3^1 x 5^1 x 83^1

Step 2: Now we found out the prime factors of 1245. The second step is to make pairs of those prime factors. Since 1245 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 1245 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 1245, we need to group it as 45 and 12.

Step 2: Now we need to find n whose square is 12. We can say n as ‘3’ because 3 x 3 is lesser than or equal to 12. Now the quotient is 3, and after subtracting 9 from 12, the remainder is 3.

Step 3: Now let us bring down 45 which is the new dividend. Add the old divisor with the same number, 3 + 3, and we get 6 which will be our new divisor.

Step 4: The new divisor will be 62n. We need to find the value of n.

Step 5: The next step is finding 62n x n ≤ 345. Let us consider n as 5. Now 625 x 5 = 3125.

Step 6: Subtract 3125 from 3450. The difference is 325, and the quotient is 35.

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 32500.

Step 8: Now we need to find the new divisor that is 705 because 705 x 5 = 3525.

Step 9: Subtracting 3525 from 32500 we get the result 29275.

Step 10: Now the quotient is 35.26

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

So the square root of √1245 is approximately 35.268.

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 1245 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1245. The smallest perfect square less than 1245 is 1225, and the largest perfect square greater than 1245 is 1296. √1245 falls somewhere between 35 and 36.

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 (1245 - 1225) ÷ (1296 - 1225) = 20 / 71 ≈ 0.282 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 35 + 0.282 ≈ 35.268, so the square root of 1245 is approximately 35.268.

• 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 a 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: It is the expression of a number as the product of its prime factors. For example, the prime factorization of 18 is 2 x 3 x 3.
   
• Approximation method: A technique used to find an estimated value of a number, often used for calculating non-perfect square roots.