Question

What is the simplest radical form of $\sqrt{8}$?

Answer 1

Hint: We solve this problem by taking out the possible perfect square out of the square root to get its simplest form. The simplest form is nothing but the representation of a given number making the value inside the square root its least value.
We take the number 8 and represent it in the form of a product of one perfect square and some other number so that we can take out the perfect square out of square root to get the required answer.

Complete step-by-step solution:
We are asked to find the simplest radical form of $\sqrt{8}$
Let us assume that the required value as,
$\Rightarrow x=\sqrt{8}$
We know that the simplest radical form of a given number is the representation of a given number making the value inside the square root its least value by taking the perfect square out of the square root.
We know that, in order to represent the given number in simplest form we need to take the given number as the product of perfect square and some other number.
We know that there are only 2 perfect squares under 8 that are 1 and 4.
Here, we can see that the representation of the product of 1 is the same as the number.
So, let us represent the number 8 as product of 4 then we get,
$\Rightarrow x=\sqrt{4\times 2}$
We know that the standard formula of square roots that is,
$\Rightarrow \sqrt{a\times b}=\sqrt{a}\times \sqrt{b}$
By using this formula in above equation then we get,
$\Rightarrow x=\sqrt{4}\times \sqrt{2}$
We know that the value of square root of 4 is 2 that is $\sqrt{4}=2$
By using this result in above equation then we get,
$\begin{align}
  & \Rightarrow x=2\times \sqrt{2} \\
 & \Rightarrow x=2\sqrt{2} \\
\end{align}$
Here, we can see that we cannot represent the number 2 as a product of perfect squares and some other number.
So, we can conclude that the simplest radical form of given number $\sqrt{8}$ is $2\sqrt{2}$ that is,
$\therefore \sqrt{8}=2\sqrt{2}$

Note: We are asked to find the simplest radical form of $\sqrt{8}$ but not the value of $\sqrt{8}$
So, we need to represent the number having the value inside the square root such that it cannot be represented as a product of perfect square and some other number by taking out the possible perfect square out of square root.But some students may make mistakes finding the value of square root which is not the required answer.