Question 1162018
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Your original equation is


    {{{log(root(3,x^2+1))}}} = {{{1/3}}}.


It means that


    {{{root(3,x^2+1)}}} = 10^(1/3).


Cube both sides.  You will get then


    {{{x^2 + 1}}} = 10,


which implies


    x^2 = 10 - 1 = 9


    x = +/- {{{sqrt(9)}}} = +/- 3.


<U>ANSWER</U>.  The given equation has 2 (two) solutions:  x= 3  and  x= -3.


You can  <U>CHECK</U>  it on your own that both values satisfy the original equation.
For it, simply substitute the values into the equation.
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Solved.