Question

What is the square root of $768$ in simplified 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 we can see that $768$ is not a perfect square. To solve this we factorize the given number.

Complete step-by-step solution:
Given, $\sqrt {768} $
$768$ can be factored as,
$768 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3$
We can see that $2$ is multiplied eight times, so we write in exponential form and raise $2$ to the power $8$. So, we get,
$ \Rightarrow 768 = {2^8} \times 3$
Now, $\sqrt {768} = \sqrt {{2^8} \times 3} $
We know that ${2^8} = {\left( {{2^4}} \right)^2}$.
Since we know that ${2^8}$ is a perfect square. So, we can take this outside of the square root we have,
So, $\sqrt {768} = {2^4}\sqrt 3 $
Also, we know that the value of ${2^4} = 16$.
Since $3$ is not perfect square, we can multiply this and keep it inside the square root,
$ \Rightarrow \sqrt {768} = 16\sqrt 3 $
This is the simplified radical form of $\sqrt {768} $.

Note: Here $\sqrt{}$ is the radical symbol used to represent the root of numbers. 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.