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Root sum or integer? (Posted on 2013-08-18) Difficulty: 3 of 5
Determine which of these is larger:
       29√14 + 4√15  or,    124
*** For an extra challenge, solve this problem using only pen and paper.

No Solution Yet Submitted by K Sengupta    
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the hard way | Comment 1 of 6

I'm sure there is a much more elegant solution than this one as it requires a lot of multiplications by hand, however it was accomplished with just a couple pieces of paper.
since both values are positive we can square both and compare the result.  Which we get
12014+232*sqrt(210) and 15376
subtracting 12014 from both values does not affect the comparison thus giving us
232*sqrt(210) and 3362
again squaring both sides does not affect the comparison thus we get
11303040 and 11303044
since 11303044>11303040 we can conclude that
124>29*sqrt(14)+4*sqrt(15)
this can be confirmed with a calculator
29*sqrt(14)+4*sqrt(15)=123.9999976....

of interesting note, as its extreme proximity caused me to wonder.  Of all the numbers of the form a*sqrt(b)+c*sqrt(d)  with a,b,c,d being a positive integer of 1 or 2 digits and b,c being squarefree, the given number is the single closest to 124.  However I have a feeling this is no coincidence.


  Posted by Daniel on 2013-08-18 11:18:11
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