Square Root of 227

The square root is the inverse of the square of the number. 227 is not a perfect square. The square root of 227 is expressed in both radical and exponential form. In radical form, it is expressed as √227, whereas (227)^(1/2) in exponential form. √227 ≈ 15.0665, 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 227 is broken down into its prime factors.

Step 1: Finding the prime factors of 227 227 is a prime number, so it cannot be broken down into smaller prime factors. Since 227 is not a perfect square, calculating √227 using prime factorization 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 227, we need to group it as 27 and 2.

Step 2: Now we need to find n whose square is ≤ 2. We can say n as ‘1’ because 1 × 1 is lesser than or equal to 2. Now the quotient is 1, after subtracting 1 from 2, the remainder is 1.

Step 3: Now let us bring down 27, which is the new dividend. Add the old divisor with the same number 1 + 1, we get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 127. Let's consider n as 5, now 25 × 5 = 125.

Step 6: Subtract 127 from 125, the difference is 2, and the quotient is 15.

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 200.

Step 8: Now we need to find the new divisor, which is 301 because 301 × 0 = 0.

Step 9: Subtracting 0 from 200, we get the result 200, and we continue the process.

Step 10: Now the quotient is 15.0

Step 11: Continue doing these steps until we get two numbers after the decimal point or until the remainder is zero.

So the square root of √227 is approximately 15.066.

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 227 using the approximation method.

Step 1: Now we have to find the closest perfect square of √227. The smallest perfect square less than 227 is 225, and the largest perfect square greater than 227 is 256. √227 falls somewhere between 15 and 16.

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 (227 - 225) / (256 - 225) = 2 / 31 ≈ 0.065. 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 15 + 0.065 ≈ 15.065, so the square root of 227 is approximately 15.065.

• 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 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 number: A prime number is a number greater than 1 that has no divisors other than 1 and itself. Example: 227 is a prime number.
   
• Decimal approximation: When the square root of a number is not a perfect square, its decimal approximation is used to represent its value. Example: √227 ≈ 15.0665.