Square root of 168
2026-02-28 23:21 Diff
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Last updated on August 5, 2025

If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in the field of vehicle design, finance etc. Here, we will discuss the square root of 168.

What is the 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.

Finding the square root of 168

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 
     

Square Root of 168 by Prime Factorization 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.
 

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Square Root of 168 by Long Division Method

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

Square root of 168 by approximation method

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
 

Common Mistakes and How to Avoid Them in the Square Root of 168

Students do make mistakes while finding the square root, likewise forgetting about the negative square root. Skipping long division methods etc. Now let us look at a few of those mistakes that students tend to make in detail.
 

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Problem 1

A square-shaped room is 168 square feet, calculate the length of each side of the room.

Okay, lets begin

The length of each side of the room is 12.96 square feet
 

Explanation

 To calculate the length of the side of the room, we need to calculate the square root of 168

√168  = 12.96

This tells us the length of the side of the square room is 12.96 square feet
 

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Problem 2

A square shaped building measuring 168 square feet is built; if each sides is √168 , what will be the square feet of half of the building?

Okay, lets begin

 84 square meters
 

Explanation

We can just divide given area by 2 as the building is square shaped


Dividing 168 by 2 = we get 84

So half of the building measures 84 square meters
 

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Problem 3

Calculate √168 x √5?

Okay, lets begin

64.8
 

Explanation

 The first step is to find the square root of 168 which is 12.96, the second step is to multiply 12.96 with 5

So 12.96 x 5  = 64.8
 

Well explained 👍

Problem 4

√168 Can this be expressed as a fraction?

Okay, lets begin

Yes, it can be expressed as a fraction

Explanation

we can express the square root of 168 as a fraction, as it is a irrational number

Well explained 👍

Problem 5

Identify the length of the side of the square table if the area is 168 square feet.

Okay, lets begin

12.96 square feet
 

Explanation

To calculate the side of the table we need to identify the square root of 168 that is 168 = 12.96

Well explained 👍

FAQ on square root of 168

1.What is √168 in its simplest form?

 The prime factorization of 168 is 2 x  2 x 2 x 3 x 7 so the simplest form of √168  = √(2 x 2 x 2 x 3 x 7) 

2.Mention the factors of 168.

Factors of 168 are  1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, and 168

3.Calculate the square of 168.

We get the square of 168 by multiplying the number by itself, that is 168 x 168 = 28224
 

4.Is 168 a prime number?

5.168 is divisible by?

168 has many factors; those are 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, and 168.

Important Glossaries for the Square Root of 168

  • 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

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

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