Square Root of 166

The square root is the inverse of the square of the number. 166 is not a perfect square. The square root of 166 is expressed in both radical and exponential form. In the radical form, it is expressed as √166, whereas (166)^(1/2) in the exponential form. √166 ≈ 12.8841, 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 166 is broken down into its prime factors.

Step 1: Finding the prime factors of 166 Breaking it down, we get 2 x 83: 2¹ x 83¹

Step 2: Now we found out the prime factors of 166. Since 166 is not a perfect square, the digits of the number cannot be grouped in pairs.

Therefore, calculating √166 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 166, we need to group it as 66 and 1.

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

Step 3: Now let us bring down 66, 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 x n ≤ 66. Let us consider n as 2, now 2 x 2 x 2 = 44

Step 6: Subtract 66 from 44; the difference is 22, and the quotient is 12.

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 2200.

Step 8: Now we need to find the new divisor that is 9 because 249 x 9 = 2241

Step 9: Subtracting 2241 from 2200 we get the result -41.

Step 10: Now the quotient is 12.9

Step 11: 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 √166 is approximately 12.88.

The approximation method is another method for finding the 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 166 using the approximation method.

Step 1: Now we have to find the closest perfect square of √166. The smallest perfect square of 166 is 144 and the largest perfect square of 166 is 169. √166 falls somewhere between 12 and 13.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula (166 - 144) / (169 - 144) = 0.88 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 12 + 0.88 = 12.88.

So the square root of 166 is approximately 12.88.

• 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, 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, but it is usually the positive square root that is used due to its relevance in real-world applications. This is known as the principal square root.
   
• Perfect square: A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it is 4^2.
   
• Decimal: A decimal is a number that includes a whole number and a fractional part, separated by a decimal point. For example, 7.86, 8.65, and 9.42 are decimals.