<s>[INST] Simplify $\sqrt n+\frac {1}{\sqrt n}$ for $n=7+4\sqrt3$ If $n=7+4\sqrt3$,then what is the simplified value of 
$$\sqrt n+\frac {1}{\sqrt n}$$
I was taking LCM but how to get rid of  $\sqrt n$ in denominator
 [/INST] You could as OP suggests, add 2, and take the square root.  
For example, $\dfrac{1}{(7+4\sqrt3)}$ is $(7-4 \sqrt3)$, so the sum of these numbers is $14$.  You add $2$ to it to get $16$, and take the square root.  
The new number is $4 = x + \dfrac {1}x$,  which leads to $x = \dfrac{\sqrt{4+2} + \sqrt{4-2}}{2}$ = 1.93185165259.
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