Square Root of 444

The square root is the inverse of the square of a number. 444 is not a perfect square. The square root of 444 is expressed in both radical and exponential form. In the radical form, it is expressed as √444, whereas (444)^(1/2) in the exponential form. √444 ≈ 21.0713, 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 444 is broken down into its prime factors.

Step 1: Finding the prime factors of 444 Breaking it down, we get 2 × 2 × 3 × 37: 2² × 3¹ × 37¹

Step 2: Now we found out the prime factors of 444. The second step is to make pairs of those prime factors. Since 444 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 444 using prime factorization is impossible.

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 444, we need to group it as 44 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 × 2 = 4. Now the quotient is 2, and after subtracting 4 - 4, the remainder is 0.

Step 3: Now let us bring down 44, 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 ≤ 44. Let us consider n as 1, now 4 × 1 = 4, which is less than 44.

Step 6: Subtract 44 from 40, the difference is 4, and the quotient is 21.

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 400.

Step 8: Now we need to find the new divisor that is 421. We need to find n such that 421n × n ≤ 400. Let's consider n as 0, because 421 × 0 = 0.

Step 9: Subtracting 0 from 400 we get the result 400.

Step 10: Continuing these steps, we can approximate the square root of 444 to 21.07.

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 444 using the approximation method.

Step 1: Now we have to find the closest perfect square to √444. The smallest perfect square less than 444 is 400, and the largest perfect square more than 444 is 441. √444 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). Using the formula (444 - 400) / (441 - 400) = 44/41 ≈ 1.073. Adding this to the nearest whole number, we get 21 + 0.073 = 21.073.

So the square root of 444 is approximately 21.073.

• 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, it is usually the positive square root that is used in real-world applications, known as the principal square root.

• Prime factorization: The expression of a number as a product of its prime factors.

• Long division method: A technique used to find the square root of a number by dividing it into smaller parts.