Question

How do you find the square root of the $16$?

Answer 1

Hint: In this problem we need to calculate the square root of the given number. For this we are going to use the prime factorization method. In which we are going to find all the prime factors of the given number. For this we will check whether the given number is divisible by prime numbers like $2$, $3$, $5$, $7$, and so on. After that we will use the exponential formula $a.a.a.a.a...\text{ n times}={{a}^{n}}$ and write the given number in exponential form. Now we will apply the square root function which is nothing but the ${{\left( \dfrac{1}{2} \right)}^{th}}$ power of the obtained value and simplify the equation to get the required result.

Complete step-by-step solution:
Given number is $16$.
Checking whether the number $16$ is divisible by $2$ or not. We can say that the number $16$ is divisible by $2$ and gives $8$ as a result. Mathematically we can write this as
$16=2\times 8$
Considering the number $8$. Checking whether the number $8$ is divisible by $2$ or not. We can say that the number $8$ is divisible by $2$ and gives $4$ as a result. Mathematically we can write this as
$8=2\times 4$
Considering the number $4$. Checking whether the number $4$ is divisible by $2$ or not. We can say that the number $4$ is divisible by $2$ and gives $2$ as a result. Mathematically we can write this as
$4=2\times 2$
From all these values, we can write
$\begin{align}
  & 16=2\times 8 \\
 & \Rightarrow 16=2\times 2\times 4 \\
 & \Rightarrow 16=2\times 2\times 2\times 2 \\
\end{align}$
Applying the formula $a.a.a.a.a...\text{ n times}={{a}^{n}}$ in the above equation, then we will get
$16={{2}^{4}}$
Applying the square root function on both sides of the above equation, then we will have
$\sqrt{16}=\sqrt{{{2}^{4}}}$
We know that the square root is the ${{\left( \dfrac{1}{2} \right)}^{th}}$ power of the number, so the above equation is modified as
$\sqrt{16}={{\left( {{2}^{4}} \right)}^{\dfrac{1}{2}}}$
Applying the exponential rule ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$ in the above equation, then we will get
$\begin{align}
  & \sqrt{16}={{2}^{4\times \dfrac{1}{2}}} \\
 & \Rightarrow \sqrt{16}={{2}^{2}} \\
\end{align}$
We know that the value of ${{2}^{2}}$ is $4$. From this we can write
$\therefore \sqrt{16}=4$.

Note: We can see that for a small number we have too many checks in the prime factorization method. For a big number it will be more difficult to check the prime factors. So, we can use another method which is a long division method to find the square root of the number.