Square Root of 1240

The square root is the inverse of the square of the number. 1240 is not a perfect square. The square root of 1240 is expressed in both radical and exponential form. In radical form, it is expressed as √1240, whereas (1240)^(1/2) in exponential form. √1240 ≈ 35.2136, 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 1240 is broken down into its prime factors.

Step 1: Finding the prime factors of 1240 Breaking it down, we get 2 × 2 × 2 × 5 × 31: 2^3 × 5^1 × 31^1

Step 2: Now we found out the prime factors of 1240. The second step is to make pairs of those prime factors. Since 1240 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1240 using prime factorization is not possible to yield a perfect square root.

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 1240, we need to group it as 40 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 12 - 9, the remainder is 3.

Step 3: Now let us bring down 40, 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 6n. Now, find the value of n.

Step 5: The next step is finding 6n × n ≤ 340. Let us consider n as 5, now 65 × 5 = 325.

Step 6: Subtract 340 from 325; the difference is 15, and the quotient is 35.

Step 7: Since the remainder is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to bring down pairs of zeros. Now the new dividend is 1500.

Step 8: Now we need to find the next digit for the divisor. Consider 705 × 2 = 1410.

Step 9: Subtracting 1410 from 1500, we get the result 90.

Step 10: Now the quotient is 35.2.

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 √1240 is approximately 35.21.

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 1240 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1240. The smallest perfect square less than 1240 is 1225, and the largest perfect square greater than 1240 is 1296. √1240 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 (1240 - 1225) ÷ (1296 - 1225) = 15 ÷ 71 ≈ 0.2113 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 + 0.2113 ≈ 35.2113, so the square root of 1240 is approximately 35.21.

• 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 a principal square root.
   
• Prime factorization: It is the process of expressing a number as the product of its prime factors. For example, the prime factorization of 1240 is 2 × 2 × 2 × 5 × 31.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.