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Subject:

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