Square Root of 3121

The square root is the inverse of the square of the number. 3121 is not a perfect square. The square root of 3121 is expressed in both radical and exponential form. In the radical form, it is expressed as √3121, whereas (3121)^(1/2) in the exponential form. √3121 ≈ 55.834, 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 3121 is broken down into its prime factors:

Step 1: Finding the prime factors of 3121 Breaking it down, we get 3121 = 7 × 11 × 41

Step 2: Now we found out the prime factors of 3121. The second step is to make pairs of those prime factors. Since 3121 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 3121 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 3121, we need to group it as 21 and 31.

Step 2: Now we need to find n whose square is less than or equal to 31. We can say n is 5 because 5 × 5 = 25 which is less than 31. Now the quotient is 5, and after subtracting 25 from 31, the remainder is 6.

Step 3: Bring down 21 which is the new dividend. Add the old divisor with the same number 5 + 5, we get 10 which will be our new divisor.

Step 4: The new divisor will be the sum of the previous quotient and divisor. Now we get 10n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 10n × n ≤ 621. Let us consider n as 5, now 105 × 5 = 525.

Step 6: Subtract 621 from 525, the difference is 96, and the quotient is 55.

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 zeros to the dividend. Now the new dividend is 9600.

Step 8: Now we need to find the new divisor that is 558 because 558 × 8 = 4464.

Step 9: Subtracting 4464 from 9600 we get the result 5136.

Step 10: Now the quotient is 55.8.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero. So the square root of √3121 is approximately 55.83.

The approximation method is another method for finding the 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 3121 using the approximation method.

Step 1: Now we have to find the closest perfect square of √3121. The smallest perfect square less than 3121 is 3025 (55^2) and the largest perfect square greater than 3121 is 3136 (56^2). √3121 falls somewhere between 55 and 56.

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 (3121 - 3025) / (3136 - 3025) = 96 / 111 ≈ 0.865. 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 55 + 0.865 = 55.865, so the square root of 3121 is approximately 55.865.

• Square root: A square root is the inverse of a square. Example: 5^2 = 25 and the inverse of the square is the square root that is √25 = 5.

• 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.

• 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.

• Perfect square: A perfect square is an integer that is the square of an integer. For example, 1, 4, 9, 16, and 25 are perfect squares.