Square Root of 406

The square root is the inverse of the square of the number. 406 is not a perfect square. The square root of 406 is expressed in both radical and exponential form. In radical form, it is expressed as √406, whereas (406)^(1/2) in the exponential form. √406 ≈ 20.1246, 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 406 is broken down into its prime factors.

Step 1: Finding the prime factors of 406 Breaking it down, we get 2 x 203, where 203 is further broken down to 7 x 29. Therefore, the prime factors of 406 are 2 x 7 x 29.

Step 2: Since 406 is not a perfect square, we cannot form pairs of the prime factors.

Therefore, calculating 406 using prime factorization to find an exact square root is not feasible.

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 406, we need to group it as 06 and 4.

Step 2: Now we need to find n whose square is less than or equal to 4. We can say n as ‘2’ because 2 x 2 = 4. Now the quotient is 2, and after subtracting 4 - 4, the remainder is 0.

Step 3: Bring down 06, 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 4n. We need to find the value of n such that 4n x n ≤ 06.

Step 5: Considering n as 1, we get 41 x 1 = 41.

Step 6: Subtract 41 from 60, and the difference is 19. The quotient is 20.

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 1900.

Step 8: We need to find the new divisor. Let us consider n as 4; then 404 x 4 = 1616.

Step 9: Subtracting 1616 from 1900, we get the result 284.

Step 10: Now the quotient is 20.1. 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 √406 is approximately 20.12.

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 406 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √406. The smallest perfect square less than 406 is 400 (20^2), and the largest perfect square greater than 406 is 441 (21^2). √406 falls somewhere between 20 and 21.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (406 - 400) / (441 - 400) = 6/41 ≈ 0.146. 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 20 + 0.146 ≈ 20.146.

So the approximate square root of 406 is 20.146.

• Square root: A square root is the inverse of a square. Example: 4^2 = 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: This is the process of expressing a number as a product of its prime numbers.

• Long division method: A method used to find the square roots of non-perfect squares through repeated division and subtraction steps.