Square root of 168

The square root is the inverse of the square of the number. 168 is not a perfect square. The square root of 168 is expressed in both radical and exponential form. In the radical form it is expressed as, √168 ,  whereas (168)^1/2 in the exponential form. √168  = 12.96148, 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

• 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 168 is broken down into its prime factors 

Step 1: Finding the prime factors of 168

Breaking it down, we get 2 x 2 x 2 x 3 x 7:  2³  x 3¹  x 7¹

Step 2: Now we found out the prime factors of 168. The second step is to make pairs of those prime factors. Since 168 is not a perfect square, therefore the digits of the number can’t be grouped in pair. Therefore, calculating 168  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 168, we need to group it as 68 and 1. 

Step 2: Now we need to find n whose square is 1. We can say n as ‘1’ because 1 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 68 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 2 n × n ≤ 68 let us consider n as 2, now 22 x  2 = 44

Step 6: Subtract 68 from 44 the difference is 24, 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 2400.

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

Step 9: Subtracting 2241 from 2400 we get the result 159.

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 is no decimal values continue till the remainder is zero

So the square root of √168  is 12. 96

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 168  using the approximation method

Step 1: Now we have to find the closest perfect square of √168 

The smallest perfect square of 168 is 144 and the largest perfect square of 168 is 169. √168  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 (168 - 144)  ÷ (169-144) = 0.96

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 + 0.96 = 12.96, so the square root of 168 is 12.96

12 + 0.96 = 12.96, so the square root of 168 is 12.96
 

• Square root: A square root is the Inverse of a square .Example: 4² =16   and the inverse of square is square root that is  √ 16 = 4

• Irrational number: An irrational number is a number that cannot be written in the form of p/q, 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 a positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.

• Integers: The combination of whole numbers and negative numbers are integers for example: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6,7………….are integers

• Decimal: If a number has a whole number and a fraction in a single number then it is called a decimal for example: 7.86, 8.65, and 9.42 are decimals