Square Root of 1206

The square root is the inverse of the square of the number. 1206 is not a perfect square. The square root of 1206 is expressed in both radical and exponential form. In radical form, it is expressed as √1206, whereas in exponential form, it is (1206)^(1/2). √1206 ≈ 34.742, 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 1206 is broken down into its prime factors:

Step 1: Finding the prime factors of 1206 Breaking it down, we get 2 x 3 x 3 x 67: 2^1 x 3^2 x 67^1

Step 2: Now we found out the prime factors of 1206. The second step is to make pairs of those prime factors. Since 1206 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1206 using prime factorization is not straightforward.

The long division method is particularly used for non-perfect square numbers. This method involves finding 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 1206, we need to group it as 06 and 12.

Step 2: Now we need to find n whose square is ≤ 12. We can say n is 3 because 3 x 3 = 9, which is less 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 06, which is the new dividend. Add the old divisor with the same number 3 + 3 to get 6, which will be our new divisor.

Step 4: The new divisor will be 6n, where we need to find the value of n such that 6n x n ≤ 306.

Step 5: Let us consider n as 5, now 65 x 5 = 325, which is too large, so we try n = 4.

Step 6: Subtract 264 (64 x 4) from 306, the difference is 42, 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 4200.

Step 8: Find the new divisor, which is 689 because 689 x 6 = 4134.

Step 9: Subtracting 4134 from 4200, we get the result 66.

Step 10: Now the quotient is approximately 34.7.

Step 11: Continue doing these steps until we get two numbers after the decimal point.

So the square root of √1206 is approximately 34.74.

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 1206 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1206. The smallest perfect square less than 1206 is 1156, and the largest perfect square greater than 1206 is 1225. √1206 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 (1206 - 1156) / (1225 - 1156) ≈ 0.742. 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 + 0.742 = 34.742, so the square root of 1206 is approximately 34.742.

• 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.
   
• 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 are numbers that divide a given number without leaving a remainder. For example, factors of 12 are 1, 2, 3, 4, 6, and 12.
   
• Long division method: A method used to find the square root of non-perfect squares by systematically dividing and finding a quotient that brings the remainder close to zero.