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Code:

sage: var( 'f,g' );
sage: LHS = - f/3/ ( -g/2 + 1/6*sqrt( 4/3*f^3 + 9*g^2) )^(1/3)
sage: RHS =        ( -g/2 - 1/6*sqrt( 4/3*f^3 + 9*g^2) )^(1/3)
sage: bool( LHS == RHS )
False
sage: bool( LHS^3 == RHS^3 )
True

Comments:

If a bool on a to be equality is answered by sage with False, this means that the equality is mathematically either false (for instance if we mistype a g/2 as g), or true, but sage could not prove it in the latter case. In case sage gives us the True, then it is really true in mathematics. In our case, sage is giving us the True only after passing to the third power. (And i would do the same mathematically, since it depends of the implicit choice of the third root for the to places to make the choice. If one correlates the roots, this is ok. But where should sage or i know this?)

Code:

sage: var( 'f,g' );
sage: LHS = - f/3/ ( -g/2 + 1/6*sqrt( 4/3*f^3 + 9*g^2) )^(1/3)
sage: RHS =        ( -g/2 - 1/6*sqrt( 4/3*f^3 + 9*g^2) )^(1/3)
sage: bool( LHS == RHS )
False
sage: bool( LHS^3 == RHS^3 )
True

Comments:

If a bool on a to be equality is answered by sage with False, this means that the equality is mathematically either false (for instance if we mistype a g/2 as g), or a plus as a minus), or true, but sage could not prove it in the latter case. In case sage gives us the True, then it is really true in mathematics. In our case, sage is giving us the True only after passing to the third power. (And i would do the same mathematically, since it depends of the implicit choice of the third root for the to places to make the choice. If one correlates the roots, this is ok. But where should sage or i know this?)

Code:

sage: var( 'f,g' );
sage: LHS = - f/3/ ( -g/2 + 1/6*sqrt( 4/3*f^3 + 9*g^2) )^(1/3)
sage: RHS =        ( -g/2 - 1/6*sqrt( 4/3*f^3 + 9*g^2) )^(1/3)
sage: bool( LHS == RHS )
False
sage: bool( LHS^3 == RHS^3 )
True

Comments:

If a bool on a to be equality is answered by sage with False, this means that the equality is mathematically either false (for instance if we mistype a g/2 as g or a plus as a minus), minus in the formulas for LHS or RHS above), or true, but sage could not prove it in the latter case. In case sage gives us the True, then it is really true in mathematics. In our case, sage is giving us the True only after passing to the third power. (And i would do the same mathematically, since it depends of the implicit choice of the third root for the to places to make the choice. If one correlates the roots, this is ok. But where should sage or i know this?)

Code:

sage: var( 'f,g' );
sage: LHS = - f/3/ ( -g/2 + 1/6*sqrt( 4/3*f^3 + 9*g^2) )^(1/3)
sage: RHS =        ( -g/2 - 1/6*sqrt( 4/3*f^3 + 9*g^2) )^(1/3)
sage: bool( LHS == RHS )
False
sage: bool( LHS^3 == RHS^3 )
True

Comments:

If a bool on a to be equality is answered by sage with False, this means that the equality is mathematically either false (for instance if we mistype a g/2 as g or a plus as a minus in the formulas for LHS or RHS above), or true, but sage could not prove it in the latter case. In case sage gives us the True, then it is really true in mathematics. In our case, sage is giving us the True only after passing to the third power. (And i would do the same mathematically, since without doing this it depends of the implicit choice of the third root for the to places to make the choice. If one correlates the roots, this is ok. But where should sage or i know this?)

Code:

sage: var( 'f,g' );
sage: LHS = - f/3/ ( -g/2 + 1/6*sqrt( 4/3*f^3 + 9*g^2) )^(1/3)
sage: RHS =        ( -g/2 - 1/6*sqrt( 4/3*f^3 + 9*g^2) )^(1/3)
sage: bool( LHS == RHS )
False
sage: bool( LHS^3 == RHS^3 )
True

Comments:

If a bool on a to be equality is answered by sage with False, this means that the equality is mathematically either false (for instance if we mistype a g/2 as g or a plus as a minus in the formulas for LHS or RHS above), or true, but sage could not prove it in the latter case. In case sage gives us the True, then it is really true in mathematics. In our case, sage is giving us the True only after passing to the third power. (And i would do the same mathematically, since without doing this it depends of on the implicit choice of the third root for the to places to make the choice. If one correlates the roots, this is ok. But where should sage or i know this?)

Code:

sage: var( 'f,g' );
sage: LHS = - f/3/ ( -g/2 + 1/6*sqrt( 4/3*f^3 + 9*g^2) )^(1/3)
sage: RHS =        ( -g/2 - 1/6*sqrt( 4/3*f^3 + 9*g^2) )^(1/3)
sage: bool( LHS == RHS )
False
sage: bool( LHS^3 == RHS^3 )
True

Comments:

If a bool on a to be equality is answered by sage with False, this means that the equality is mathematically either false (for instance if we mistype a g/2 as g or a plus as a minus in the formulas for LHS or RHS above), or true, but sage could not prove it in the latter case. In case sage gives us the True, then it is really true in mathematics. In our case, sage is giving us the True only after passing to the third power. (And i would do the same mathematically, since without doing this it depends on the implicit choice of the third root for the to two places to make the choice. If one correlates the roots, this is ok. But where should sage or i know this?)