Square Root of 1242

The square root is the inverse of the square of a number. 1242 is not a perfect square. The square root of 1242 is expressed in both radical and exponential form. In radical form, it is expressed as √1242, whereas in exponential form it is (1242)^(1/2). √1242 ≈ 35.2293, 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, for non-perfect square numbers, 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 1242 is broken down into its prime factors:

Step 1: Finding the prime factors of 1242 Breaking it down, we get 2 x 3 x 3 x 3 x 23: 2^1 x 3^3 x 23^1

Step 2: Now we have found the prime factors of 1242. The next step is to make pairs of those prime factors. Since 1242 is not a perfect square, the digits of the number can’t be grouped into pairs perfectly.

Therefore, calculating √1242 using prime factorization is not straightforward.

The long division method is particularly used for non-perfect square numbers. In this method, we check for the closest perfect square 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 digits of 1242 from right to left. Grouping results in 12 and 42.

Step 2: Now we need to find a number whose square is close to or less than 12. The number is 3 because 3 squared is 9. This gives a quotient of 3, and subtracting 9 from 12 leaves a remainder of 3.

Step 3: Bring down 42 to make the new dividend 342. Double the current quotient to get the new divisor's base: 2 × 3 = 6.

Step 4: Find a digit 'n' such that 6n × n ≤ 342. By trial, 65 × 5 = 325, which is less than 342.

Step 5: Subtract 325 from 342, leaving a remainder of 17.

Step 6: Since the dividend is less than the divisor, add a decimal point to the quotient and bring down two zeros to the remainder, making it 1700.

Step 7: The new divisor base is 70 (from 65), and find a digit 'n' such that 70n × n ≤ 1700.

Step 8: Continue this process to get more decimal places.

So, the square root of √1242 is approximately 35.229.

The approximation method is another way to find square roots. It is an easy method to find the square root of a given number. Let's learn how to find the square root of 1242 using the approximation method.

Step 1: Identify the closest perfect squares around 1242. The closest perfect squares are 1225 (35^2) and 1296 (36^2). Thus, √1242 falls between 35 and 36.

Step 2: Use the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). Using the formula: (1242 - 1225) / (1296 - 1225) = 17 / 71 ≈ 0.239 Adding this to 35, the approximate square root of 1242 is 35 + 0.239 = 35.239

• Square root: A square root is the inverse of squaring a number. For example, 4^2 = 16, and the inverse of the square is the square root, so √16 = 4.
   
• Irrational number: An irrational number cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Perfect square: A number that is the square of an integer. For example, 36 is a perfect square because it can be written as 6^2.
   
• Prime factorization: The process of expressing a number as the product of its prime factors.
   
• Decimal: A number that includes a whole number and a fractional part, represented with a decimal point, such as 7.86, 8.65, and 9.42.