let a,b,c,k be rational number such that k is not a perfect cube if a+6k1/3 +ck2/3=0 then prove that a=b=c=0.

Asked by  | 13th Jul, 2014, 08:06: PM

Expert Answer:

G i v e n space t h a t space a plus b k to the power of 1 third end exponent plus c k to the power of 2 over 3 end exponent equals 0.... left parenthesis 1 right parenthesis N o w space m u l t i p l y space t h e space a b o v e space e q u a t i o n space b y space k to the power of 1 third end exponent comma space w e space h a v e k to the power of 1 third end exponent open parentheses a plus b k to the power of 1 third end exponent plus c k to the power of 2 over 3 end exponent close parentheses equals 0 rightwards double arrow a k to the power of 1 third end exponent plus b k to the power of 2 over 3 end exponent plus c k equals 0.... left parenthesis 2 right parenthesis M u t l p l y space t h e space e q u a t i o n space left parenthesis 1 right parenthesis space b y space b space a n d space t h e space e q u a t i o n space left parenthesis 2 right parenthesis space b y space c comma space w e space h a v e a b plus b squared k to the power of 1 third end exponent plus b c k to the power of 2 over 3 end exponent equals 0.... left parenthesis 3 right parenthesis a c k to the power of 1 third end exponent plus b c k to the power of 2 over 3 end exponent plus c squared k equals 0... left parenthesis 4 right parenthesis S u b t r a c t space t h e space e q u a t i o n space left parenthesis 4 right parenthesis space f r o m space left parenthesis 3 right parenthesis comma space w e space h a v e comma open parentheses b squared minus a c close parentheses k to the power of 1 third end exponent plus a b minus c squared k equals 0 S i n c e space k space i s space i r r a t i o n a l comma space w e space h a v e comma b squared minus a c equals 0 space a n d space a b minus c squared k equals 0 rightwards double arrow b squared equals a c..... a n d space a b equals c squared k.... left parenthesis 5 right parenthesis

S q u a r i n g space e q u a t i o n space left parenthesis 5 right parenthesis comma space w e space h a v e a squared b squared equals c to the power of 4 k squared rightwards double arrow a squared open parentheses a squared close parentheses equals c to the power of 4 k squared rightwards double arrow a squared open parentheses a c close parentheses equals c to the power of 4 k squared rightwards double arrow a cubed c minus c to the power of 4 k squared equals 0 rightwards double arrow c open parentheses a cubed minus k squared c cubed close parentheses equals 0 rightwards double arrow c equals 0 space o r space a cubed minus k squared c cubed equals 0 rightwards double arrow c equals 0 space o r space a cubed equals k squared c cubed rightwards double arrow c equals 0 space o r fraction numerator space a cubed over denominator c cubed end fraction equals k squared rightwards double arrow c equals 0 space o r space k squared equals open parentheses a over c close parentheses cubed rightwards double arrow c equals 0 space o r space k to the power of 2 over 3 end exponent equals open parentheses a over c close parentheses B u t space k space i s space i r r a t i o n a l space a n d space h e n c e space w e space h a v e comma space a cubed minus k squared c cubed not equal to 0 rightwards double arrow c equals 0 S i n c e space b squared equals a c comma space w e space h a v e comma space b equals 0 b equals 0 space a n d space c equals 0 space a n d space f r o m space e q u a t i o n space left parenthesis 1 right parenthesis comma space w e space h a v e comma space a equals 0 T h u s comma space w e space h a v e comma space a equals 0 comma space b equals 0 space a n d space c equals 0

Answered by Vimala Ramamurthy | 14th Jul, 2014, 09:31: AM