Question

Find the square root of the number $216$.

Answer 1

Hint: For this problem we need to calculate the square root of the given number. In this problem we are going to use the prime factorization method. In this method we will write the given number as multiples of the prime numbers like $2$, $3$, $5$, $7$, $11$ and so on. For this we will check whether the given number is divisible by the prime numbers or not. After that we will write the given number as multiple of the prime numbers. Now we will use the exponential rule which is $a.a.a.a.a....\text{ n times}={{a}^{n}}$ and simplify the equation. In order to calculate the square root, we will apply square root function to the both sides of the above equation.

Complete step-by-step solution:
Given number is $216$.
Checking whether the number is divisible by $2$ or not. We can say that the given number is divisible by $2$ and give $108$ as quotient. So, we are going to write the given number as
$\Rightarrow 216=2\times 108$
Considering the number $108$. Checking whether the number $108$ is divisible by $2$ or not. We can say that the number $108$ is divisible by $2$ and gives $54$ as the quotient. So, we are going to write the number $108$ as
$\Rightarrow 108=2\times 54$
Considering the number $54$ . Checking whether the number $54$ is divisible by $2$ or not. We can say that the number $54$ is divisible by $2$ and gives $27$ as the quotient. So, we are going to write the number $54$ as
$\Rightarrow 54=2\times 27$.
Considering the number $27$. Checking whether the number $27$ is divisible by $2$ or not. We can see that the number $27$ is not divisible by $2$. So, moving to the next prime number which is $3$. Checking whether the number $27$ is divisible by $3$ or not. We can say that the number $27$ is divisible by $3$ and gives $9$ as a result. So, we are going to write the number $27$ as
$\Rightarrow 27=3\times 9$
We know that the value $9$ can be written as $9=3\times 3$.
From all the above values we are going to write the given number as
$\begin{align}
  & \Rightarrow 216=2\times 108 \\
 & \Rightarrow 216=2\times 2\times 54 \\
 & \Rightarrow 216=2\times 2\times 2\times 27 \\
 & \Rightarrow 216=2\times 2\times 2\times 3\times 9 \\
 & \Rightarrow 216=2\times 2\times \left( 2\times 3 \right)\times 3\times 3 \\
\end{align}$
Applying the exponential rule $a.a.a.a.a....\text{ n times}={{a}^{n}}$ in the above equation, then we will get
$\Rightarrow 216={{2}^{2}}\times {{3}^{2}}\times 6$
Applying the square root on both sides of the above equation, then we will have
$\begin{align}
  & \Rightarrow \sqrt{216}=\sqrt{{{2}^{2}}\times {{3}^{2}}\times 6} \\
 & \Rightarrow \sqrt{216}=2\times 3\sqrt{6} \\
 & \Rightarrow \sqrt{216}=6\sqrt{6} \\
\end{align}$
Hence the square root of the given number $216$ is $6\sqrt{6}$.

Note: While writing the given number in exponential form we should take that there are maximum possible factors that have power as multiples of $2$. Because we know that the square root is the inverse function of the square. So, when we apply them at a time, they both get cancelled to each other and simplifies our solution.