Cube Root Calculation, ExplainedDate: 04/18/2002 at 10:22:01 From: Harsha Ppriya Subramanya Subject: Cube root manually Dear Dr. Math, It was good to see the way you outlined to calculate the cube root manually, but I wasn't able to understand. I would appreciate it if you could show the same for a small number like 8 (whose cube root we know to be 2) and for a number like 9. This would help me and any student to understand the method clearly. Regards, Harsha Priya Subramanya Date: 04/18/2002 at 12:56:56 From: Doctor Peterson Subject: Re: Cube root manually Hi, Harsha. Which method did you see and not understand? We describe several methods in the links here: Square/Cube Roots Without a Calculator - Dr. Math FAQ http://mathforum.org/dr.math/faq/faq.sqrt.by.hand.html#cube The method shown in Cube roots - Dr. Math archives http://mathforum.org/library/drmath/view/52606.html uses 8 as an example. This is the easiest method to use and remember. If you want the other method demonstrated again, let me know. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 04/18/2002 at 14:08:44 From: Harsha Ppriya Subramanya Subject: Cube root manually Dear Dr. Math, This is the link that I need some explanation on. Probably if you could illustrate the same for a number like 8 or 9 it would be helpful. Cube Root by Hand http://mathforum.org/library/drmath/view/52605.html This method looks good only if it could be understood. Regards, Harsha Priya Subramanya Date: 04/18/2002 at 14:45:13 From: Doctor Peterson Subject: Re: Cube root manually Hi, Harsha. As I suggested, this is a more complex method, hard to remember and requiring a lot of guessing, so the other method is probably preferred, but it does have its advantages where humans have to do all the work, simply because you never have to divide by a long decimal. Let's take the cube root of 8: 3 _2.________ \/ 8.000 000 8 - 0 The first step is to "Find the largest cube of a single-digit integer less than it"; this is 2^3=8, so we write 2 as the answer, and subtract 8, giving a remainder of 0. We're done! Okay, that was too easy. Let's try 9 this time: 3 _2.________ \/ 9.000 000 8 - 1 000 I've brought down the next group of three digits, and we have to find the next digit in the answer. This will be the digit x for which (300a^2 + 30ax + x^2)x is less than 1000. (I'm calling the answer so far, 2, by the name "a".) To do that, we note that 300a^2 = 300*2^2 = 1200, and 30a = 30*2 = 60. We then write (1200+60*_+_^2)*_=1000 Well, since 1200 is already too big to divide 1000, this digit has to be 0. We continue: 3 _2.__0_____ \/ 9.000 000 8 - 1 000 0 ----- 1 000 000 This time a = 20 (we ignore the decimal point), so 300a^2 = 120000 and 30a = 600, so we write (120,000+600*_+_^2)_ = 1,000,000 For a first guess, as suggested at the bottom of the page, we can just divide 1,000,000 by 120,000, which gives x = 8; then (120,000+600*8+8^2)8 = 124,864*8 = 998,912 which does the job quite nicely. So now we have 3 _2.__0___8_ \/ 9.000 000 8 - 1 000 0 ----- 1 000 000 998 912 --------- 1 088 I'll let you continue if you wish; the correct cube root is about 2.080083823. For comparison, the other method works like this: First guess: 2 Second guess: 2/3 * 2 + 1/3 * 9/2^2 = 1.33333 + 0.75 = 2.0833 Third guess: 2/3 * 2.0833 + 1/3 * 9/2.0833^2 = 1.38886 + 0.69122 = 2.08008 We've already got five significant digits correct. The other method, of course, guarantees that every digit we've calculated is correct, and probably took less actual calculation if you take into account every big square and division I had to do this way; so it's not too bad, if you can remember it all. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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