Square Root of 618

The square root is the inverse of the square of a number. 618 is not a perfect square. The square root of 618 is expressed in both radical and exponential forms. In the radical form, it is expressed as √618, whereas in the exponential form it is expressed as (618)^(1/2). √618 ≈ 24.849, 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 618 is broken down into its prime factors.

Step 1: Finding the prime factors of 618 Breaking it down, we get 2 x 3 x 103.

Step 2: Now we found out the prime factors of 618. The second step is to make pairs of those prime factors. Since 618 is not a perfect square, calculating 618 using prime factorization is not straightforward.

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 618, we need to group it as 18 and 6.

Step 2: Now we need to find n whose square is 6. We can say n as ‘2’ because 2 x 2 = 4 is lesser than or equal to 6. Now the quotient is 2 after subtracting 6-4, the remainder is 2.

Step 3: Now let us bring down 18, which is the new dividend. Add the old divisor with the same number 2 + 2 to get 4, which will be our new divisor.

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

Step 5: The next step is finding 4n × n ≤ 218, let us consider n as 5, now 4 x 5 x 5 = 100.

Step 6: Subtract 218 from 100, the difference is 118, and the quotient is 24.

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 11800.

Step 8: Now we need to find the new divisor that is 496 because 496 × 2 = 992.

Step 9: Subtracting 992 from 11800, we get the result 10008.

Step 10: Now the quotient is 24.8.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero. So the square root of √618 is approximately 24.849.

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 618 using the approximation method.

Step 1: Now we have to find the closest perfect square of √618. The smallest perfect square less than 618 is 576, and the largest perfect square greater than 618 is 625. √618 falls somewhere between 24 and 25.

Step 2: Now we need to apply the formula that is (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). Going by the formula (618 - 576) ÷ (625 - 576) = 42 ÷ 49 = 0.857. 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 24 + 0.857 = 24.857, so the square root of 618 is approximately 24.857.

• 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, the positive square root is often used due to its applications in the real world. That is why it is also known as the principal square root.

• Prime factorization: The process of breaking down a number into its prime factors. For example, 618 = 2 × 3 × 103.

• Approximation method: A method to estimate the square root of non-perfect squares by finding the nearest perfect squares and using a formula to calculate a close approximation.