Square Root of 447

The square root is the inverse of the square of the number. 447 is not a perfect square. The square root of 447 is expressed in both radical and exponential form. In the radical form, it is expressed as √447, whereas (447)^(1/2) in the exponential form. √447 ≈ 21.166, 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 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 447 is broken down into its prime factors:

Step 1: Finding the prime factors of 447 Breaking it down, we get 3 x 149, where 149 is a prime number.

Step 2: Now we found out the prime factors of 447. Since 447 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 447 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 447, we need to group it as 47 and 4.

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

Step 3: Now let us bring down 47, 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 4n. We need to find the value of n such that 4n x n ≤ 47. Let us consider n as 1, now 41 x 1 = 41.

Step 5: Subtract 47 from 41, the difference is 6, and the quotient is 21.

Step 6: 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 600.

Step 7: Now we need to find the new divisor, which is 42 (since the quotient is 21, double it and add 1). So 421 x 1 = 421.

Step 8: Subtracting 421 from 600 we get the result 179.

Step 9: 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 √447 is approximately 21.16.

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 447 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √447. The smallest perfect square less than 447 is 441 (21^2) and the largest perfect square more than 447 is 484 (22^2). √447 falls somewhere between 21 and 22.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square)

Applying the formula: (447 - 441) / (484 - 441) = 6 / 43 ≈ 0.14 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 21 + 0.14 = 21.14, so the square root of 447 is approximately 21.14.

• 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, 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. Example: 9 is a perfect square because it is 3^2.

• Long division method: A method used to find the square root of non-perfect squares by division, which involves grouping and repeated subtraction.

• Decimal approximation: The process of finding a decimal value close to the exact square root of a non-perfect square for practical uses.