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If  a1/3+b1/3+c1/3=0then show that (a+b+c)3=27abc

Asked by pannalal.malakar | 29th May, 2014, 09:35: AM

Expert Answer:

G i v e n space t h a t space a to the power of 1 third end exponent plus b to the power of 1 third end exponent plus c to the power of 1 third end exponent equals 0. W e space n e e d space t o space p r o v e space t h a t space open parentheses a plus b plus c close parentheses cubed equals 27 a b c. W e space k n o w space t h a t space A cubed plus B cubed plus C cubed minus 3 A B C equals open parentheses A plus B plus C close parentheses open parentheses A squared plus B squared plus C squared minus A B minus B C minus C A close parentheses.... left parenthesis 1 right parenthesis I n space t h i s space p r o b l e m comma space w e space h a v e comma space A equals a to the power of 1 third end exponent comma space B equals b to the power of 1 third end exponent comma space C equals c to the power of 1 third end exponent. T h u s comma 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 open parentheses a to the power of 1 third end exponent close parentheses cubed plus open parentheses b to the power of 1 third end exponent close parentheses cubed plus open parentheses c to the power of 1 third end exponent close parentheses cubed minus 3 open parentheses a to the power of 1 third end exponent close parentheses open parentheses b to the power of 1 third end exponent close parentheses open parentheses c to the power of 1 third end exponent close parentheses equals open parentheses space a to the power of 1 third end exponent plus b to the power of 1 third end exponent plus c to the power of 1 third end exponent close parentheses open curly brackets open parentheses a to the power of 1 third end exponent close parentheses squared plus open parentheses b to the power of 1 third end exponent close parentheses plus open parentheses c to the power of 1 third end exponent close parentheses squared minus open parentheses a to the power of 1 third end exponent close parentheses open parentheses b to the power of 1 third end exponent close parentheses minus open parentheses b to the power of 1 third end exponent close parentheses open parentheses c to the power of 1 third end exponent close parentheses minus 3 open parentheses a to the power of 1 third end exponent close parentheses open parentheses c to the power of 1 third end exponent close parentheses close curly brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 0 cross times open curly brackets open parentheses a to the power of 1 third end exponent close parentheses squared plus open parentheses b to the power of 1 third end exponent close parentheses plus open parentheses c to the power of 1 third end exponent close parentheses squared minus open parentheses a to the power of 1 third end exponent close parentheses open parentheses b to the power of 1 third end exponent close parentheses minus open parentheses b to the power of 1 third end exponent close parentheses open parentheses c to the power of 1 third end exponent close parentheses minus 3 open parentheses a to the power of 1 third end exponent close parentheses open parentheses c to the power of 1 third end exponent close parentheses close curly brackets equals 0 space space space open square brackets S i n c e space a to the power of 1 third end exponent plus b to the power of 1 third end exponent plus c to the power of 1 third end exponent equals 0 close square brackets rightwards double arrow open parentheses a to the power of 1 third end exponent close parentheses cubed plus open parentheses b to the power of 1 third end exponent close parentheses cubed plus open parentheses c to the power of 1 third end exponent close parentheses cubed equals 3 open parentheses a to the power of 1 third end exponent close parentheses open parentheses b to the power of 1 third end exponent close parentheses open parentheses c to the power of 1 third end exponent close parentheses rightwards double arrow a plus b plus c equals 3 open parentheses a to the power of 1 third end exponent close parentheses open parentheses b to the power of 1 third end exponent close parentheses open parentheses c to the power of 1 third end exponent close parentheses C u b i n g space o n space b o t h space t h e space s i d e s comma space w e space h a v e comma open parentheses a plus b plus c close parentheses cubed equals 3 cubed open curly brackets open parentheses a to the power of 1 third end exponent close parentheses open parentheses b to the power of 1 third end exponent close parentheses open parentheses c to the power of 1 third end exponent close parentheses close curly brackets cubed rightwards double arrow open parentheses a plus b plus c close parentheses cubed equals 27 a b c

Answered by Vimala Ramamurthy | 29th May, 2014, 10:33: AM