Question

How do you simplify the square root of $95$?

Answer 1

Hint:
To calculate the square root student should always find out the factors of the given number. After simplifying the number in terms of its prime factors, factors that appear twice should be taken out of the square root and should be written once. In this method the student should know the factorization method to find the factors of the given number. Factorization method is important not only for finding the square root but also for finding the LCM of a given number. The last step is the repetition of the second step i.e. taking out all the common multiples and writing them once and leaving the remaining factors inside the square root.

Complete step by step solution:
First step is to find out the factors of $95$ by factorization method.
After factorization we get the following factors of $95$
\[\sqrt {95} = \sqrt {19 \times 5} ......(1)\]
We can see from the above step that none of the numbers can be further simplified as both of them are prime numbers itself. Thus the square root of \[95\] is $\sqrt {95} $ itself.
Since this contains no square factors, √95 is already in simplest form. There are no factors that can be moved outside the radical.
As a continued fraction, we find:

$\sqrt {95} = \{ 9;\overline {1,2,1,18} \} $=$9 + \dfrac{1}{{1 + \dfrac{1}{{2 + \dfrac{1}{{1 + \dfrac{1}{{18 + \dfrac{1}{{1 + \dfrac{1}{{2 + \dfrac{1}{{1 + \dfrac{1}{{18 + ......}}}}}}}}}}}}}}}}.......(1)$

Hence an approx and an efficient rational approximation for $\sqrt {95} $ is:
$\{ 9;\overline {1,2,1} \} $=$9 + \dfrac{1}{{1 + \dfrac{1}{{2 + \dfrac{1}{{1 + .....}}}}}}...........(2)$

Solving the above equation, we see that the simplified answer for $\sqrt {95} $ is $\dfrac{{39}}{4}$$ \cong 9.75$.

Note:
The student should learn this method if he/she is not acquainted with it. This method is very useful in competitive exams and also for fast and approximate calculation of the square root of prime numbers is to be found out and also when the calculator is not available. This simplification method needs a lot of practice and understanding.