If x+y+z = 0, find fraction numerator x squared y squared space plus space y squared z squared space plus space z squared x squared over denominator x to the power of 4 space plus space y to the power of 4 space end exponent plus space z to the power of 4 end fraction

Asked by a.behera67 | 5th Nov, 2015, 08:02: PM

Expert Answer:

G i v e n space t h a t space x plus y plus z equals 0 W e space k n o w space t h a t comma x to the power of 4 plus y to the power of 4 plus z to the power of 4 equals open parentheses x plus y plus z close parentheses open parentheses x cubed plus y cubed plus z cubed close parentheses minus open parentheses x y plus y z plus z x close parentheses open parentheses x squared plus y squared plus z squared close parentheses plus x y z open parentheses x plus y plus z close parentheses space space space space space space space space space space space space space space space space space equals open parentheses x plus y plus z close parentheses open parentheses x cubed plus y cubed plus z cubed close parentheses minus open parentheses x y plus y z plus z x close parentheses open square brackets open parentheses x plus y plus z close parentheses squared minus 2 x y minus 2 y z minus 2 x z close square brackets plus x y z open parentheses x plus y plus z close parentheses space space space space space space space space space space space space space space space space space equals 0 cross times open parentheses x cubed plus y cubed plus z cubed close parentheses minus open parentheses x y plus y z plus z x close parentheses open square brackets 0 squared minus 2 x y minus 2 y z minus 2 x z close square brackets plus x y z cross times 0 space space space space space space space open square brackets because x plus y plus z equals 0 close square brackets space space space space space space space space space space space space space space space space space equals negative open parentheses x y plus y z plus z x close parentheses open square brackets negative 2 x y minus 2 y z minus 2 x z close square brackets space space space space space space space space space space space space space space space space space space space equals 2 open parentheses x y plus y z plus z x close parentheses open square brackets x y plus y z plus x z close square brackets space space space space space space space space space space space space space space space space space equals 2 open parentheses x y plus y z plus z x close parentheses squared.... left parenthesis 1 right parenthesis R e c i p r o c a t i n g space t h e 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 fraction numerator 1 over denominator x to the power of 4 plus y to the power of 4 plus z to the power of 4 end fraction equals fraction numerator 1 over denominator 2 open parentheses x y plus y z plus z x close parentheses squared end fraction M u l t i p l y i n g space t h e space n u m e r a t o r comma space x squared y squared plus y squared z squared plus z squared x squared space o n space b o t h space t h e space s i d e s space w e space h a v e comma fraction numerator space x squared y squared plus y squared z squared plus z squared x squared over denominator x to the power of 4 plus y to the power of 4 plus z to the power of 4 end fraction equals fraction numerator space x squared y squared plus y squared z squared plus z squared x squared over denominator 2 open parentheses x y plus y z plus z x close parentheses squared end fraction... left parenthesis 2 right parenthesis A g a i n space c o n s i d e r comma space t h e space n u m e r a t o r space o f space t h e space r i g h t space h a n d space s i d e. x squared y squared plus y squared z squared plus z squared x squared equals open parentheses x y plus y z plus z x close parentheses squared minus 2 x y z open parentheses x plus y plus z close parentheses... left parenthesis 3 right parenthesis T h u s space s u b s t i t u t i n g space t h e space v a l u e space o f space space f r o m space e q u a t i o n space left parenthesis 3 right parenthesis space i n space e q u a t i o n space left parenthesis 2 right parenthesis comma space w e space h a v e comma fraction numerator space x squared y squared plus y squared z squared plus z squared x squared over denominator x to the power of 4 plus y to the power of 4 plus z to the power of 4 end fraction equals fraction numerator open parentheses x y plus y z plus z x close parentheses squared minus 2 x y z open parentheses x plus y plus z close parentheses over denominator 2 open parentheses x y plus y z plus z x close parentheses squared end fraction 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 fraction numerator open parentheses x y plus y z plus z x close parentheses squared minus 2 x y z cross times 0 over denominator 2 open parentheses x y plus y z plus z x close parentheses squared end fraction space space space space open square brackets because x plus y plus z equals 0 close square 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 equals fraction numerator open parentheses x y plus y z plus z x close parentheses squared over denominator 2 open parentheses x y plus y z plus z x close parentheses squared end fraction 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 1 half

Answered by Vimala Ramamurthy | 6th Nov, 2015, 10:46: AM

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