Assume that the cube has a side length r, equal on all sides.

Surface area = r² on each side, for 6r² total.

The diagonal of the cube is sqrt (r² + r² + r²) = sqrt (3r²) = r sqrt 3

The square of the diagonal is then r² sqrt(3)² = 3r²

Twice the diagonal is then 6r², equal to the surface area.

The volume of a cube is r³.

The cube of its diagonal is r³ sqrt(3)³ = 3r³sqrt(3)

1/9 cube root of 3 = (3-²)(3^(1/3)) = 3^(-5/3)

1/9 cube root of 3 times the cube of the diagonal is 3^(-5/3) * 3 * sqrt(3) * r³

Which simplifies out to 3^(-1/6) * r³, or the volume times the sixth root of 3.

HOWEVER:

If you instead say that the volume of a cube is 1/9 SQUARE root of 3, times the cube of the diagonal,

1/9 sqrt(3) = 3-² * 3^(1/2) = 3^(-3/2)

3^(-3/2) * 3r³sqrt(3)

= 3^(-3/2) * 3 * 3^(1/2) * r³

= 3^0 * r³ = 1*r³ = r³.

So this is probably what your problem was asking for in the first place.