Question 881765
First simplify



{{{root(3,81) = root(3,27*3)}}}



{{{root(3,81) = root(3,27)*root(3,3)}}}



{{{root(3,81) = 3*root(3,3)}}}



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Then take the reciprocal. Make sure to rationalize the denominator.



{{{1/(root(3,81)) = 1/(3*root(3,3))}}}



{{{1/(root(3,81)) = (1*red(root(3,3)*root(3,3)))/(3*root(3,3)*red(root(3,3)*root(3,3)))}}} Multiply top and bottom by 2 copies of {{{root(3,3)}}} to  rationalize the denominator.



{{{1/(root(3,81)) = (root(3,3*3))/(3*root(3,3*3*3))}}}



{{{1/(root(3,81)) = (root(3,9))/(3*root(3,27))}}}



{{{1/(root(3,81)) = (root(3,9))/(3*3)}}}



{{{1/(root(3,81)) = (root(3,9))/(9)}}}



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So the reciprocal of {{{root(3,81)}}} is {{{(root(3,9))/(9)}}}