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2011 Powered Digits (Posted on 2011-03-25) Difficulty: 3 of 5
Find the first digit before and after the decimal point in the base ten expansion of (√2 + √3)2011.

For an extra challenge, solve this puzzle without using a computer program.

No Solution Yet Submitted by K Sengupta    
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The Broll guesses... | Comment 1 of 5

To avoid annoying powers, let 2^(1/2)=a, 3^(1/2)=b

Checking small even values:

(a+b)^6=  969.998969.
(a+b)^8=  9,601.99989
(a+b)^10= 95,049.999989...
(a+b)^12= 940,897.9999989...
(a+b)^14= 9,313,929.99999989....
(a+b)^16= 92,198,401.999999989
(a+b)^20= 9,034,502,497.999999999889....

and a pattern seems to be emerging.

Now (a+b)2011

= (a+b)2000 *(a+b)^11

=(something huge)+0.9999999999........9*(a+b)^11

(a+b)^11=299052.4282999780776

so the digits before and after the decimal point should be ...2.4...

Edited on March 25, 2011, 11:36 am
  Posted by broll on 2011-03-25 11:30:42

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