Question

What is $ \sqrt {192} $ in simplest radical form?

Answer 1

Hint: Square root of a number is a value, which on multiplied by itself gives the original number. Suppose, ‘x’ is the square root of ‘y’, then it is represented as \[x = \sqrt y \] or we can express the same equation as \[{x^2} = y\]. Here \[\sqrt {} \] is the radical symbol used to represent the root of numbers.

Complete step-by-step answer:
Take the given expression: $ \sqrt {192} $
Find the factors of the term inside the square root.
Prime factorization is the process of finding which prime numbers can be multiplied together to make the original number, where prime numbers are the numbers greater than $ 1 $ and which are not the product of any two smaller natural numbers.
 $ \sqrt {192} = \sqrt {2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3} $
It can be re-written as a square of terms- Also, applied the fundamental principle that the square and square-root cancel each other.
 $
  \sqrt {192} = \sqrt {\underline {2 \times 2} \times \underline {2 \times 2} \times \underline {2 \times 2} \times 3} \\
  \sqrt {192} = \sqrt {{2^2} \times {2^2} \times {2^2} \times 3} \\
  \sqrt {192} = \sqrt {{2^2} \times {2^2} \times {2^2}} \times \sqrt 3 \\
  \sqrt {192} = 2 \times 2 \times 2 \times \sqrt 3 \\
  \sqrt {192} = 8\sqrt 3 \;
  $
This is the required solution.
So, the correct answer is “ $ 8\sqrt 3 $ ”.

Note: The number under the radical symbol is called radicand. The positive number, when multiplied by itself, represents the square of the number. The square root of the square of a positive number gives the original number. To find the factors, find the smallest prime number that divides the given number and divide it by that number, and then again find the smallest prime number that divides the number obtained and so on. The set of prime numbers obtained that are multiplied to each other to form the bigger number are called the factors.