Date: Mon, 21 Sep 1992 19:46:44 -0400 (EDT)
From: Matthew John Bushey <mb8d+@andrew.cmu.edu>

Does anyone out there know what is the cubed root of 81?

Just wondering....

Well, the "root of 81" is 9 (recall that when you don't say
what kind of root you want, the default is "square"), and
9 cubed is 729.

... Eh? Oh, you meant the "cube root", not the "cubed root"?
Well, that's another kettle of fish entirely. The n'th root
of x is equal to x raised to the power 1/n. I fed this to
my friendly Common Lisp system:

> (expt 81 1/3)
4.3267487109222245

If I were you, I wouldn't trust the last few digits of this
approximation, but fifteen decimal places ought to hold you
for now.

Here's how you could estimate it in your head.
Note that 81 = 3 to the fourth power, so

  1/3      4  1/3    4/3        1/3
81    = ( 3  )    = 3    = 3 ( 3    )

Now, the cube root of 3 is surely between 1 and 2,
because 1 cubed is 1 and 2 cubed is 8.  So the cube
root of 3 is 1 plus some smaller fractional amount x.
               3                2    3
So  3 = (1 + x)  = 1 + 3 x + 3 x  + x     (binomial expansion).

                  3
Let's ignore the x  term, which is probably small because
x is sort of small.  Then

             2                   2
1 + 3 x + 3 x  = 3     so   x + x  = 2/3 .

                            2
Hm... if x = 1/2, then x + x  = 3/4, which is a bit
                                               2
too big.  So figure x is about 0.4; then  x + x  = .4 + .16 = .56
which is too small.  So probably x is about 0,45 or so.

So the cube root of 3 is about 1.45, and the cube root of
81 is 3 times that, or about 4.35 -- not a bad approximation.

--Guy STeele