$$\eqalign{
& \root 3 \of {26 - 15\sqrt 3 } + \root 3 \of {26 + 15\sqrt 3 } \cr
& = \root 3 \of {8 + 18 - 12\sqrt 3 - 3\sqrt 3 } + \root 3 \of {8 + 18 + 12\sqrt 3 + 3\sqrt 3 } \cr
& = \root 3 \of {{2^3} - {{3.2}^2}\sqrt 3 + 3.2.{{\left( {\sqrt 3 } \right)}^2} - {{\left( {\sqrt 3 } \right)}^3}} + \root 3 \of {{2^3} + {{3.2}^2}\sqrt 3 + 3.2.{{\left( {\sqrt 3 } \right)}^2} + {{\left( {\sqrt 3 } \right)}^3}} \cr
& = \root 3 \of {{{\left( {2 - \sqrt 3 } \right)}^3}} + \root 3 \of {{{\left( {2 + \sqrt 3 } \right)}^3}} \cr
& = 2 - \sqrt 3 + 2 + \sqrt 3 = 4 \cr} $$
