The ratio of the side of two similar cubes is 3: 4 the smaller cube has a volume of 729 what is the volume of the larger cube

Lemme see, notice, this is the relationship from side ratios, 
to areas and volumes
keep in mind that areas are square figures, involving 2 units,
and volumes are cubic figures, involving 3 units
thus 

what the dickens all that means?

well, you have two cubes, both similar, their ratio, is 3:4,
3 is smaller than 4, thus is from smaller to bigger cube, 3:4 ratio

the smaller cube has a volume, or  of 729 cubic units,
what's the other volume?

well, let us use those proportions above

solve for "v"

notice, the numerator in 3:4, has the smaller volume, 729,
the bigger is at the bottom of that proportion

Just divide the x axis (the top) by 2 or multiply the y axis (the bottom) by 2