Question

Find the square root of the number 7921

Answer 1

Hint: Find the prime factorization of the given number. Then take the square root at both the sides to get the required square root of the number.

Complete step-by-step solution -
In the question, we have to find the square root of the number 7921.
So, here we will find the prime factorization of the number 7921.
So we will get:
\[\begin{align}
  & \Rightarrow 7921=89\times 89 \\
 & \Rightarrow 7921={{89}^{2}} \\
\end{align}\]
Now, the square root of 7921 is written as \[\sqrt{7921}\]
So here we will take the square root as both the sides of the \[7921={{89}^{2}}\] to get the square root \[\sqrt{7921}\].
Now, taking the square root both sides in above equation, we get:
\[\Rightarrow \sqrt{7921}=\sqrt{{{89}^{2}}}\]
Now, it is just required to further simplify the above expression. So, we will do that as follows:
\[\begin{align}
  & \Rightarrow \sqrt{7921}=\sqrt{{{89}^{2}}} \\
& \Rightarrow \sqrt{7921}={{({{89}^{2}})}^{\dfrac{1}{2}}}\,\,\,\,\,\,\,\,\,\because \sqrt{b}={{(b)}^{\dfrac{1}{2}}}\, \\
 & \Rightarrow \sqrt{7921}=(89)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\because \,{{({{a}^{2}})}^{\dfrac{1}{2}}}=a \\
\end{align}\]

So finally, the square root of 7921 is 89.

Note: There is a difference between the prime factorization of the number and finding the factors of the number. The product of prime factorization is the number itself but the product of the factors of the number will be greater than the number itself. For example the factors of 10 are 1,2,5,10 but the prime factorization of 10 is as follows:\[10=2\times 5\].
Alternative method to solve the question is:
Find the prime factorization of the given number. All the powers of the prime factors are then divided by two, to get the factors which are then multiplied together to get the square root.