What to the third power is equal to 729

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.         .  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.)

The correct answer is $90