Prove that :(i) left parenthesis 2 to the power of 30 space end exponent plus 2 to the power of 29 plus 2 to the power of 28 right parenthesis obelus divided by left parenthesis 2 to the power of 31 plus 2 to the power of 30 minus 2 to the power of 29 right parenthesis equals 7 over 10
                (ii)open parentheses x to the power of a over x to the power of b close parentheses to the power of a squared plus a b plus b squared end exponent cross times space open parentheses x to the power of b over x to the power of c close parentheses to the power of b squared plus b c plus c squared end exponent cross times open parentheses x to the power of c over x to the power of a close parentheses to the power of c squared plus c a plus a squared end exponent equals 1  

Asked by araima2001 | 5th Sep, 2014, 03:33: PM

Expert Answer:

open parentheses I close parentheses C o n s i d e r space t h e space l e f t space h a n d space s i d e space o f space t h e space g i v e n space e q u a t i o n : fraction numerator open parentheses 2 to the power of 30 plus 2 to the power of 29 plus 2 to the power of 28 close parentheses over denominator open parentheses 2 to the power of 31 plus 2 to the power of 30 minus 2 to the power of 29 close parentheses end fraction equals fraction numerator 2 to the power of 28 open parentheses 2 squared plus 2 to the power of 1 plus 2 to the power of 0 close parentheses over denominator 2 to the power of 29 open parentheses 2 squared plus 2 to the power of 1 minus 2 to the power of 0 close parentheses 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 equals fraction numerator open parentheses 4 plus 2 plus 1 close parentheses over denominator 2 open parentheses 4 plus 2 minus 1 close parentheses 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 equals fraction numerator 7 over denominator 2 cross times 5 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 equals 7 over 10 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 R i g h t space h a n d space s i d e space o f space t h e space e q u a t i o n  open parentheses i i close parentheses C o n s i d e r space t h e space l e f t h a n d space s i d e space o f space t h e space g i v e n space e q u a t i o ln : open parentheses x to the power of a over x to the power of b close parentheses to the power of a squared plus a b plus b squared end exponent cross times open parentheses x to the power of b over x to the power of c close parentheses to the power of b squared plus b c plus c squared end exponent cross times open parentheses x to the power of c over x to the power of a close parentheses to the power of c squared plus c a plus a squared end exponent equals open parentheses x to the power of a close parentheses to the power of a squared plus a b plus b squared end exponent over open parentheses x to the power of b close parentheses to the power of a squared plus a b plus b squared end exponent cross times open parentheses x to the power of b close parentheses to the power of b squared plus b c plus c squared end exponent over open parentheses x to the power of c close parentheses to the power of b squared plus b c plus c squared end exponent cross times open parentheses x to the power of c close parentheses to the power of c squared plus c a plus a squared end exponent over open parentheses x to the power of a close parentheses to the power of c squared plus c a plus a squared end exponent  equals open parentheses x to the power of a close parentheses to the power of a squared plus a b plus b squared minus c squared minus c a minus a squared end exponent cross times open parentheses x to the power of b close parentheses to the power of b squared plus b c plus c squared minus a squared minus a b minus b squared end exponent cross times open parentheses x to the power of c close parentheses to the power of c squared plus c a plus a squared minus b squared minus b c minus c squared end exponent equals open parentheses x to the power of a close parentheses to the power of a b plus b squared minus c squared minus c a end exponent cross times open parentheses x to the power of b close parentheses to the power of b c plus c squared minus a squared minus a b end exponent cross times open parentheses x to the power of c close parentheses to the power of c a plus a squared minus b squared minus b c end exponent equals equals open parentheses x to the power of a close parentheses to the power of a b plus b squared minus c squared minus c a end exponent cross times open parentheses x to the power of b close parentheses to the power of b c plus c squared minus a squared minus a b end exponent cross times open parentheses x to the power of c close parentheses to the power of c a plus a squared minus b squared minus b c end exponent equals x to the power of a squared b plus a b squared minus a squared c minus a c squared plus b squared c plus b c squared minus a squared b minus a b squared plus c squared a plus a squared c minus b squared c minus b c squared end exponent equals x to the power of 0 equals 1

Answered by Vimala Ramamurthy | 7th Sep, 2014, 12:28: PM

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