Square Root of 432

The square root is the inverse of the square of the number. 432 is not a perfect square. The square root of 432 is expressed in both radical and exponential forms. In the radical form, it is expressed as √432, whereas (432)^(1/2) in the exponential form. √432 ≈ 20.7846, 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 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 432 is broken down into its prime factors.

Step 1: Finding the prime factors of 432 Breaking it down, we get 2 x 2 x 2 x 2 x 3 x 3 x 3: 2^4 x 3^3

Step 2: Now we found out the prime factors of 432. The second step is to make pairs of those prime factors. Since 432 is not a perfect square, therefore the digits of the number can’t be grouped perfectly into pairs. Therefore, calculating √432 using prime factorization involves taking the square root of each pair and multiplying the results.

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 432, we need to group it as 32 and 4.

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

Step 3: Now let us bring down 32, 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, and we need to find the value of n.

Step 5: The next step is finding 4n × n ≤ 32. Let us consider n as 0.5, now 4 x 0.5 x 0.5 = 1

Step 6: Subtract 32 from 1. The difference is 31, and the quotient is 20.5.

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 3100.

Step 8: Now we need to find the new divisor that is 9 because 409 × 9 = 3681.

Step 9: Subtracting 3681 from 3100, we get the result -581.

Step 10: Now the quotient is 20.78.

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 √432 is approximately 20.78.

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 432 using the approximation method.

Step 1: Now we have to find the closest perfect square of √432. The smallest perfect square of 432 is 400, and the largest perfect square of 432 is 441. √432 falls somewhere between 20 and 21.

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 (432 - 400) ÷ (441 - 400) = 0.78. 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 + 0.78 = 20.78, so the square root of 432 is approximately 20.78.

• Square root: A square root is the inverse of a square. For 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.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 16 is a perfect square because it is 4^2.

• Long division method: A method used to find the square root of a number by dividing it into groups and using division to approximate the root.

• Approximation method: A technique used to estimate the square root of a number by finding close perfect squares and using interpolation.