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Simple expression but difficult answer (Posted on 2013-05-21) Difficulty: 2 of 5
Which is larger


(21/3-1)1/3     or     (1/9)1/3 - (2/9)1/3 + (4/9)1/3 ?

No Solution Yet Submitted by Danish Ahmed Khan    
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Some Thoughts numerically (spoiler) | Comment 1 of 2

Using the default precision of UBASIC:

10   Lhs=(2^(1/3)-1)^(1/3)
20   Rhs=(1/9)^(1/3)-(2/9)^(1/3)+(4/9)^(1/3)
30   print Lhs
40   print Rhs

produces

0.6381858208606441523
0.6381858208606441534

which looks like the difference is only rounding error.

This can be confirmed by increasing the precision:

point 16
Words for fractionals 16 (Decimals for display 77)
OK
run
 0.63818582086064415301550365944406770126515754397799768342106208158057548513945

 0.63818582086064415301550365944406770126515754397799768342106208158057548513989
 
which pushes the rounding error further down the line.

It seems both formulae represent the same value.


  Posted by Charlie on 2013-05-21 13:28:37
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