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Square root of 192
Did you know that 192 is the sum of ten consecutive primes? In this lesson, we will calculate the square root of 192 by long division method. We will also go through a few solved examples and interactive questions related to the square root of 192.
 Square Root of 192: √192 = 13.8564
 Square of 192: 192^{2} = 36,864
1.  What Is the Square Root of 192? 
2.  Is Square Root of 192 Rational or Irrational? 
3.  Tips and Tricks 
4.  How to Find the Square Root of 192? 
5.  FAQs on Square Root of 192 
6.  Challenging Questions 
What Is the Square Root of 192?
The square root of 192 is the value which when multiplied by itself will give the final product as 192.
Is the Square Root of 192 Rational or Irrational?
It is not possible to find a whole number which on squaring gives 192. It can be approximately written as a square of 13.856, which is a nonrecurring and nonterminating decimal number. This indicates that 192 is not a perfect square which also proves that the square root of 192 is an irrational number.
Tips and Tricks:
 192 is not a perfect square; hence, its square root is an irrational number. The square root of any number n, which is not a perfect square, will always be an irrational number.
How to Find the Square Root of 192?
The square root of 192 can be found using the following steps:
 Step 1: Check whether the number is a perfect square or not. 192 is not a perfect square as it cannot be broken down into a product of two same numbers.
 Step 2: Once the number is checked, we can do the following:
 A number which is a perfect square can be written as √x^{2} = x.
 For a number which is not a perfect square, the square root can be found using the long division method. This number can also be written in its simplified radical form of the square root.
In this case, 192 is not a perfect square. Hence, its square root is found using the long division method. The simplified radical form of the square root of 192 is given below.
Simplified Radical Form of Square Root of 192
192 can be written as the product of 3 and 64. It is given as:
√192 = √3 × 64
3 is not a perfect square. Hence, it stays within the root sign. However, 64 can be shown as 8^{2}. Thus, the simplified radical form of the square root of 192 is 8√3.
Square Root of 192 by Long Division Method
The square root of 192 can be found using the long division method. These are the steps to be followed:
 Step 1: Starting from the right, we will pair up the digits by putting a bar above them.
 Step 2: Find a number such that when you multiply it with itself, the product is less than or equal to 192. Keeping the divisor as 23, we get the quotient as 13 and the remainder 9269 = 23.
 Step 3: Double the divisor and enter it with a blank on its right. Then assume the largest digit to replace the blank  this will become the new digit in the quotient. Now, when the new divisor will be multiplied to the new quotient, the final product will be lesser than or equal to our dividend. Finally, divide and write the remainder. Guess the largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend. Divide and write the remainder. Repeat this process to get the decimal places you want.
Hence, √192 = 13.856
Explore square roots using illustrations and interactive examples
Challenging Questions:
 How will Hailey find the square root of 192 using the long division method up to 5 decimal places?
 How will Mandy express the square root of 384 in terms of square root of 192?
Square Root of 192 Solved Examples

Example 1: Can you find the two consecutive numbers between which the value of square root of 192 lies?
Solution
The perfect squares nearest to 192 are 169 and 196. The square root of 169 is 13. The square root of 196 is 14.
Hence, these are the two numbers between which the square root of 192 lies. 
Example 2: How is the square root of 192 related to the square root of 96?
Solution
Square root of 192 can be expressed as √192 = √(2 × 96)
96 can further be expressed as, 96 = 32 × 3 = 2^{5} × 3
√96 = √(2^{5} × 3) = 2^{2} × √6 = 4√6
√192 = √(2 × 2^{5} × 3) = √(2^{6} × 3) = 2^{3} × √3 = 8√3
On dividing √192 by √96 we get, √192/√96 = 8√3/4√6 = 2/√2 = √2Hence, square root of 192 is √2 times square root of 96.
FAQs on Square Root of 192
What is the square root of 192 in the simplest form?
The square root of 192 in the simplest form is 8√3.
How do you find the square root of 192?
192 is not a perfect square. Hence, the square root of 192 can be found using long division method.
What is approximate square root of 192?
The square root of 192 is 13.8564 (approximately).
Is the square root of 192 rational or irrational?
The square root of 192 is irrational.
Is the square root of 192 a real number?
Yes, the square root of 192 is a real number.
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