Square Root of 171

The square root is the inverse of the square of the number. 171 is not a perfect square. The square root of 171 is expressed in both radical and exponential forms. In the radical form, it is expressed as √171, whereas (171)^(1/2) in exponential form. √171 ≈ 13.0767, 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 171 is broken down into its prime factors.

Step 1: Finding the prime factors of 171 Breaking it down, we get 3 x 3 x 19: 3² x 19

Step 2: Now we found out the prime factors of 171. The second step is to make pairs of those prime factors. Since 171 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 171 using prime factorization is not straightforward.

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, we need to group the numbers from right to left. In the case of 171, we need to group it as 71 and 1.

Step 2: Now we need to find n whose square is ≤ 1. We can say n is ‘1’ because 1 x 1 is lesser than or equal to 1. Now the quotient is 1, and after subtracting 1 from 1, the remainder is 0.

Step 3: Now let us bring down 71, 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 20 since we add a digit and guess n such that 2n x n is less than or equal to 71.

Step 5: The next step is finding 2n x n ≤ 71. Let us consider n as 3, now 23 x 3 = 69.

Step 6: Subtract 71 from 69, the difference is 2, and the quotient is 13.

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 that is 261 because 261 x 7 ≈ 1827.

Step 9: Subtracting 1827 from 2000, we get the result 173.

Step 10: Now the quotient is 13.0.

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 √171 ≈ 13.08.

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 171 using the approximation method.

Step 1: Now we have to find the closest perfect square to √171. The smallest perfect square less than 171 is 169, and the largest perfect square greater than 171 is 196. √171 falls somewhere between 13 and 14.

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: (171 - 169) / (196 - 169) = 2/27 ≈ 0.074. 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 13 + 0.074 ≈ 13.074.

So the square root of 171 is approximately 13.074.

• 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.
   
• Prime factorization: Breaking down a number into its basic prime number components.
   
• Decimal approximation: The decimal representation of an irrational number for practical calculations.
   
• Long division method: A technique used to find the square root of non-perfect square numbers by using division and subtraction iterations.