root5+root3. proof that it is an irrational number
 

Asked by kamleshsingh826 | 23rd Aug, 2016, 08:53: PM

Expert Answer:

begin mathsize 14px style L e t space square root of 5 plus square root of 3 space b e space r a t i o n a l space e q u a l space t o space a over b.
T h e n comma space square root of 5 plus square root of 3 equals a over b
rightwards double arrow square root of 5 equals a over b minus square root of 3
rightwards double arrow open parentheses square root of 5 close parentheses squared equals open parentheses a over b minus square root of 3 close parentheses squared left parenthesis S q u a r i n g space b o t h space s i d e s right parenthesis
rightwards double arrow 5 equals a squared over b squared minus fraction numerator 2 a square root of 3 over denominator b end fraction plus 3
rightwards double arrow 2 equals a squared over b squared minus 2 square root of 3 a over b
rightwards double arrow 2 square root of 3 a over b equals a squared over b squared minus 2
rightwards double arrow 2 square root of 3 a over b equals fraction numerator a squared minus 2 b squared over denominator b squared end fraction
rightwards double arrow square root of 3 equals fraction numerator a squared minus 2 b squared over denominator 2 a b end fraction
rightwards double arrow square root of 3 space i s space r a t i o n a l comma space a space c o n t r a d i c t i o n.
H e n c e comma space square root of 5 plus square root of 3 space i s space i r r a t i o n a l.
end style

Answered by Rashmi Khot | 24th Aug, 2016, 09:36: AM