Question

How do you find \[\sqrt[2]{325}\]?

Answer 1

Hint: From the question we have been asked to find the square root of the number \[325\]. For the questions we will use the prime factorisation process and find the prime factorisation of the given number. After finding it we will use the square root form and find the required answer to the given question. So, the solution will proceed as follows.

Complete step by step solution:
Firstly, the nth root of a number x can be written generally as \[\sqrt[n]{x}\].
So, from the question we are asked to find the square root. So, when we substitute in this form our question can be written as follows.
\[\Rightarrow \sqrt[2]{325}\]
Now, we will write the prime factorisation of the number \[325\].
The prime factorisation of \[325\] will be as follows.
 \[\Rightarrow 325=5\times 5\times 13\]
We can rewrite this in terms of powers as follows.
\[\Rightarrow 325={{5}^{2}}\times 13\]
After getting this prime factorisation we will substitute this in the square root and simplify it further to get the answer. So, when we substitute it in square root we get,
\[\Rightarrow \sqrt[2]{325}\]
\[\Rightarrow \sqrt[2]{{{5}^{2}}\times 13}\]
Now, when in a square root we have a number with power two, then we can bring that number outside. So, after bringing the number five outside of the root we get,
\[\Rightarrow \sqrt[2]{{{5}^{2}}\times 13}\]
\[\Rightarrow 5\sqrt[2]{13}\]
We can remove the \[2\] on a square root as a root of a number generally represents square root. So, we get,
\[\Rightarrow 5\sqrt{13}\]
 Now, from the question we are asked to find \[\sqrt[2]{325}\]. So, we will multiply the above one with \[2\]. So, we get,
\[\Rightarrow 2\times 5\sqrt{13}\]
\[\therefore 10\sqrt{13}\]
The simplest way to find an approximation for the irrational value is to use a calculator. So, we can further simplify the answer as follows.
\[\Rightarrow 10\sqrt{13}\]
\[\Rightarrow 10\times 3.6055\]
\[\therefore 36.055\]

Note: Students must be very careful in doing the calculations. Students must have good knowledge in the concept of prime factorisation and basic operations like multiplication. Here the important point to be noted for questions of this kind is, if we are restricted to the principle square root then \[ -10\sqrt{13}\] or \[ -36.055\] is another solution.