Square Root of 6121

The square root is the inverse of the square of the number. 6121 is not a perfect square. The square root of 6121 is expressed in both radical and exponential form. In the radical form, it is expressed as √6121, whereas (6121)^(1/2) in the exponential form. √6121 ≈ 78.2279, 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 prime factorization method is not applicable; instead, 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 6121 is broken down into its prime factors:

Step 1: Finding the prime factors of 6121 Breaking it down, we get 6121 = 11 × 557.

Step 2: Now we found the prime factors of 6121. The second step is to make pairs of those prime factors. Since 6121 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 6121 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 6121, we need to group it as 61 and 21.

Step 2: Now we need to find n whose square is close to 61. We can say n is ‘7’ because 7 × 7 = 49 is lesser than 61. Now the quotient is 7, and after subtracting 49 from 61, the remainder is 12.

Step 3: Now let us bring down 21, which is the new dividend. Add the old divisor with the same number, 7 + 7, to get 14, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 14n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 14n × n ≤ 1221. Let us consider n as 8; now 148 × 8 = 1184.

Step 6: Subtract 1184 from 1221, the difference is 37, and the quotient is 78.

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 3700.

Step 8: Now we need to find the new divisor that is 785 because 785 × 5 = 3925.

Step 9: Subtracting 3925 from 3700 gives a negative result, so we try with n = 4, and 784 × 4 = 3136.

Step 10: Subtracting 3136 from 3700, we get the result 564.

Step 11: The quotient is 78.2. Step 12: 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 √6121 is approximately 78.23.

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 6121 using the approximation method.

Step 1: We have to find the closest perfect squares surrounding √6121.

The smallest perfect square less than 6121 is 6084, and the largest perfect square more than 6121 is 6400. √6121 falls somewhere between 78 and 80.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (6121 - 6084) / (6400 - 6084) = 37 / 316 ≈ 0.117

Using the formula, we identify the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 78 + 0.117 ≈ 78.12.

So the square root of 6121 is approximately 78.12.

• 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.

• Prime factorization: The process of finding which prime numbers multiply together to make the original number.

• Long division method: A method used to find the square root of numbers that are not perfect squares.