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.
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 20110325 11:30:42 