Square Root of 412

The square root is the inverse of the square of a number. 412 is not a perfect square. The square root of 412 is expressed in both radical and exponential form. In radical form, it is expressed as √412, whereas 412^(1/2) is the exponential form. √412 = 20.298, 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 long-division and approximation methods 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 412 is broken down into its prime factors.

Step 1: Finding the prime factors of 412 Breaking it down, we get 2 x 2 x 103: 2^2 x 103

Step 2: Now we found the prime factors of 412. The second step is to make pairs of those prime factors. Since 412 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 412 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 412, we need to group it as 12 and 4.

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

Step 3: Now let us bring down 12, which is the new dividend. Add the old divisor with the same number 2 + 2, we 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, and we need to find the value of n.

Step 5: The next step is finding 4n x n ≤ 12. Let us consider n as 0, and now 4 x 0 x 0 = 0.

Step 6: Subtract 12 from 0, the difference is 12, and 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 1200.

Step 8: Now we need to find the new divisor that is 204 because 204 x 5 = 1020.

Step 9: Subtracting 1020 from 1200 we get the result 180.

Step 10: Now the quotient is 20.5.

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 √412 is approximately 20.298.

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 412 using the approximation method.

Step 1: Now we have to find the closest perfect square to √412. The smallest perfect square less than 412 is 400, and the largest perfect square greater than 412 is 441. √412 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) Going by the formula (412 - 400) / (441-400) = 12 / 41 ≈ 0.293 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 + 0.293 ≈ 20.293, so the square root of 412 is approximately 20.298.

• 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. This is why it is also known as the principal square root.

• Prime factorization: A way of expressing a number as the product of its prime factors. For example, the prime factorization of 412 is 2^2 x 103.

• 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.