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Irrational Numbers

Prove that 3-...

Question

Prove that 3 - 3√7 is an irrational number

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Solution

$L e t s space a s s u m e space t h a t space 3 minus 3 square root of 7 space i s space r a t i o n a l S o space colon space space 3 minus 3 square root of 7 equals r space space space space left parenthesis space w h e r e space r space i s space a space r a t i o n a l space n u m b e r right parenthesis S q u a r i n g space o n space b o t h space s i d e s colon left parenthesis 3 minus 3 square root of 7 right parenthesis squared equals r squared space space.................... left parenthesis 1 right parenthesis A p p l y i n g space left parenthesis a minus b right parenthesis squared equals a squared minus 2 a b plus b squared S o space e q u a t i o n space left parenthesis 1 right parenthesis space b e c o m e s space colon 3 squared minus 2 cross times 3 cross times 3 square root of 7 plus open parentheses 3 square root of 7 close parentheses squared equals r squared 9 minus 18 square root of 7 plus 63 equals r squared R e a r r a n g i n g colon rightwards double arrow 18 square root of 7 equals 9 plus 63 minus r squared space space space space 18 square root of 7 equals 72 minus r squared rightwards double arrow square root of 7 equals fraction numerator 72 minus r squared over denominator 18 end fraction space space space.................. left parenthesis 2 right parenthesis H e r e space w e space c a n space s e e space square root of 7 space i s space i r r a t i o n a l space w h e r e a s space fraction numerator 72 minus r squared over denominator 18 end fraction space i s space r a t i o n a l S o space e q u a t i o n space left parenthesis 2 right parenthesis space i s space w r o n g space b e c a u s e space a n space i r r a t i o n a l space n u m b e r space c a n n o t space b e space e q u a l space t o space a space r a t i o n a l space n u m b e r. T h i s space i s space d u e space t o space t h e space w r o n g space a s s u m p t i o n space t h a t space 3 minus 3 square root of 7 space i s space r a t i o n a l S o colon rightwards double arrow 3 minus 3 square root of 7 space i s space i r r a t i o n a l H e n c e space p r o v e d$

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