If you had absolutely no idea, then you'd have roughly two choices of how to find it:
#1). Try numbers until you find the right one.
Pick a number. Cube it. If you get less than 729, go back and try a bigger number. If you get more than 729, go back and try a smaller number. Eventually you find the right one.
Try 1. 1³ = 1 . Too small. Try 2. 2³ = 8 . Still too small. Try 5. 5³ = 125 . Still too small. Try 20. 20³ = 8,000. Ooops. Too big. Try 10. 10³ = 1,000 Too big. Try 8. 8³ = 512. Oooo. Too small, but maybe getting close. Try 8.5. 8.5³ = 614.125 Still too small, but very close. Try 9. 9³ = 729 . That's it ! yay !
#2). x³ = 729
Take the log of each side: log(x³) = log(729)
3 log(x) = log(729)
Divide each side by 3 : log (x) = (1/3) log(729)
Look up log(729) : log(x) = (1/3) (2.86272...)
= 0.95424... Raise 10 to the power of each side: 10^log(x) = 10⁰·⁹⁵⁴²⁴···
x = 8.99994...
(That's the way it is with logs. They never come out even.)
