Square Root of 127

The square root is the inverse of the square of a number. 127 is not a perfect square. The square root of 127 is expressed in both radical and exponential form. In the radical form, it is expressed as √127, whereas in exponential form it is expressed as (127)^(1/2). √127 ≈ 11.2694, 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 prime factorization method is not used. Instead, 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. However, 127 is a prime number itself. Therefore, the prime factorization does not help in simplifying the square root of 127, and we consider it as √127 directly.

The long division method is particularly used for non-perfect square numbers. In this method, we 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: Group the numbers from right to left. In the case of 127, group it as 27 and 1.

Step 2: Find n whose square is less than or equal to 1. Here, n is 1 because 1 × 1 is less than or equal to 1. The quotient is 1 and the remainder is 0 after subtracting 1 - 1.

Step 3: Bring down 27 which is the new dividend. Add the old divisor with the same number, 1 + 1, to get 2 which will be our new divisor.

Step 4: Use 2 as the new divisor and find n such that 2n × n is less than or equal to 27. Let n = 5, then 2 × 5 × 5 = 50, which is greater than 27, so try n = 4, then 2 × 4 × 4 = 32, which is also greater. Use n = 3, then 2 × 3 × 3 = 18.

Step 5: Subtract 27 from 18, giving a remainder of 9, with the quotient now being 13.

Step 6: Since the dividend is less than the divisor, add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. The new dividend is 900.

Step 7: Find the new divisor, using n = 4, since 246 × 4 = 984, closer to 900.

Step 8: Subtract 984 from 900 to get -84, which means try n = 3, resulting in 243 × 3 = 729.

Step 9: Subtract 729 from 900, resulting in a remainder of 171.

Step 10: The quotient is 11.3.

Step 11: Continue these steps until you get at least two decimal places or the remainder becomes zero.

So, the approximation of the square root of 127 is 11.27.

The approximation method is another way to find square roots, and it is an easy method to find the square root of a given number. Here's how to find the square root of 127 using the approximation method:

Step 1: Find the closest perfect squares around √127. The smallest perfect square less than 127 is 121 and the largest perfect square greater than 127 is 144. Therefore, √127 falls between 11 and 12.

Step 2: Use the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using this formula: (127 - 121) / (144 - 121) = 6 / 23 ≈ 0.261. Adding this to the lower bound: 11 + 0.261 = 11.261.

Thus, the square root of 127 is approximately 11.26.

• Square root: A square root is the inverse of squaring a number. For example, 4² = 16 and the inverse of squaring is finding the square root, √16 = 4.
   
• Irrational number: An irrational number cannot be expressed in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
   
• Decimal: If a number has a whole number and a fraction in a single number, it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.
   
• Long division method: A step-by-step method used to divide large numbers that can also help find the square root of non-perfect squares.