From gls@think.com Tue Sep 22 11:50:24 1992 Received: from mail.think.com by life.ai.mit.edu (4.1/AI-4.10) id AA26659; Tue, 22 Sep 92 11:50:24 EDT Return-Path: Received: from Strident.Think.COM by mail.think.com; Tue, 22 Sep 92 11:50:22 -0400 From: Guy Steele Received: by strident.think.com (4.1/Think-1.2) id AA24984; Tue, 22 Sep 92 11:50:22 EDT Date: Tue, 22 Sep 92 11:50:22 EDT Message-Id: <9209221550.AA24984@strident.think.com> To: mb8d+@andrew.cmu.edu Cc: cube-lovers@ai.mit.edu In-Reply-To: Matthew John Bushey's message of Mon, 21 Sep 1992 19:46:44 -0400 (EDT) <0ejZvYC00WBK48jY0m@andrew.cmu.edu> Subject: cubes are great Date: Mon, 21 Sep 1992 19:46:44 -0400 (EDT) From: Matthew John Bushey 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