Square Root of 1241

The square root is the inverse of the square of the number. 1241 is not a perfect square. The square root of 1241 is expressed in both radical and exponential form. In the radical form, it is expressed as √1241, whereas (1241)^(1/2) in the exponential form. √1241 ≈ 35.2207, 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 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 1241 is broken down into its prime factors.

Step 1: Finding the prime factors of 1241 Breaking it down, we get 7 x 13 x 13: 7^1 x 13^2

Step 2: Now we found out the prime factors of 1241. The second step is to make pairs of those prime factors. Since 1241 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 1241 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 1241, we need to group it as 41 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 x 3 = 9 is lesser than or equal to 12. Now the quotient is 3. After subtracting 9 from 12, the remainder is 3.

Step 3: Now let us bring down 41, which is the new dividend. 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 the quotient. Now we get 6n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 341. Let us consider n as 5. Now 65 x 5 = 325.

Step 6: Subtract 325 from 341; the difference is 16, 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 1600.

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

Step 9: Subtracting 3525 from 1600 gives the result -1925, so adjust n to 2, making the divisor 702 x 2 = 1404.

Step 10: Subtracting 1404 from 1600 gives the result 196.

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

So the square root of √1241 is approximately 35.22.

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 1241 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1241. The smallest perfect square less than 1241 is 1225 (35^2), and the largest perfect square more than 1241 is 1296 (36^2). √1241 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 (1241 - 1225) ÷ (1296 - 1225) = 16 ÷ 71 = 0.22535 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 + 0.225 ≈ 35.225, so the square root of 1241 is approximately 35.225.

• Square root: A square root is the inverse of a square. For 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 is more prominent due to its uses in the real world. That is the reason it is also known as the principal square root.
   
• Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors.
   
• Long division method: The long division method is a way of finding the square root of a number by dividing and iterating through steps to reach an accurate decimal approximation.