if a+b+c=0 then what is the value of 1/x^b+x^-c+1 + 1/x^c+x^-a+1 + 1/x^a+x^-b+1

Asked by Nilesh | 20th Aug, 2016, 04:07: PM

Expert Answer:

begin mathsize 14px style fraction numerator 1 over denominator x to the power of b plus x to the power of negative c end exponent plus 1 end fraction plus fraction numerator 1 over denominator x to the power of c plus x to the power of negative a end exponent plus 1 end fraction plus fraction numerator 1 over denominator x to the power of a plus x to the power of negative b end exponent plus 1 end fraction
equals fraction numerator left parenthesis x to the power of c plus x to the power of negative a end exponent plus 1 right parenthesis left parenthesis x to the power of a plus x to the power of negative b end exponent plus 1 right parenthesis plus left parenthesis x to the power of b plus x to the power of negative c end exponent plus 1 right parenthesis left parenthesis x to the power of a plus x to the power of negative b end exponent plus 1 right parenthesis plus left parenthesis x to the power of b plus x to the power of negative c end exponent plus 1 right parenthesis left parenthesis x to the power of c plus x to the power of negative a end exponent plus 1 right parenthesis over denominator left parenthesis x to the power of b plus x to the power of negative c end exponent plus 1 right parenthesis left parenthesis x to the power of c plus x to the power of negative a end exponent plus 1 right parenthesis left parenthesis x to the power of a plus x to the power of negative b end exponent plus 1 right parenthesis end fraction
L e t space A equals left parenthesis x to the power of c plus x to the power of negative a end exponent plus 1 right parenthesis left parenthesis x to the power of a plus x to the power of negative b end exponent plus 1 right parenthesis
equals x to the power of c plus a end exponent plus x to the power of c minus b end exponent plus x to the power of c plus x to the power of negative a plus a end exponent plus x to the power of negative a minus b end exponent plus x to the power of negative a end exponent plus x to the power of a plus x to the power of negative b end exponent plus 1
equals x to the power of c plus a end exponent plus x to the power of c minus b end exponent plus x to the power of c plus x to the power of negative a minus b end exponent plus x to the power of negative a end exponent plus x to the power of a plus x to the power of negative b end exponent plus 2
B equals left parenthesis x to the power of b plus x to the power of negative c end exponent plus 1 right parenthesis left parenthesis x to the power of a plus x to the power of negative b end exponent plus 1 right parenthesis
equals x to the power of b plus a end exponent plus x to the power of b minus b end exponent plus x to the power of b plus x to the power of negative c plus a end exponent plus x to the power of negative c minus b end exponent plus x to the power of negative c end exponent plus x to the power of a plus x to the power of negative b end exponent plus 1
equals x to the power of a plus b end exponent plus x to the power of b plus x to the power of a minus c end exponent plus x to the power of negative c minus b end exponent plus x to the power of negative c end exponent plus x to the power of a plus x to the power of negative b end exponent plus 2
C equals left parenthesis x to the power of b plus x to the power of negative c end exponent plus 1 right parenthesis left parenthesis x to the power of c plus x to the power of negative a end exponent plus 1 right parenthesis
equals x to the power of b plus c end exponent plus x to the power of b minus a end exponent plus x to the power of b plus x to the power of negative c plus c end exponent plus x to the power of negative c minus a end exponent plus x to the power of negative c end exponent plus x to the power of c plus x to the power of negative a end exponent plus 1
equals x to the power of b plus c end exponent plus x to the power of b minus a end exponent plus x to the power of b plus x to the power of negative c minus a end exponent plus x to the power of negative c end exponent plus x to the power of c plus x to the power of negative a end exponent plus 2
A plus B plus C
equals x to the power of c plus a end exponent plus x to the power of c minus b end exponent plus x to the power of c plus x to the power of negative a minus b end exponent plus x to the power of negative a end exponent plus x to the power of a plus x to the power of negative b end exponent plus 2 space
plus x to the power of a plus b end exponent plus x to the power of b plus x to the power of a minus c end exponent plus x to the power of negative c minus b end exponent plus x to the power of negative c end exponent plus x to the power of a plus x to the power of negative b end exponent plus 2
plus x to the power of b plus c end exponent plus x to the power of b minus a end exponent plus x to the power of b plus x to the power of negative c minus a end exponent plus x to the power of negative c end exponent plus x to the power of c plus x to the power of negative a end exponent plus 2
equals x to the power of c plus a end exponent plus x to the power of c minus b end exponent plus 2 x to the power of c plus x to the power of negative a minus b end exponent plus 2 x to the power of negative a end exponent plus 2 x to the power of a plus 2 x to the power of negative b end exponent plus x to the power of a plus b end exponent plus 2 x to the power of b plus x to the power of a minus c end exponent plus x to the power of negative c minus b end exponent plus 2 x to the power of negative c end exponent
plus x to the power of b plus c end exponent plus x to the power of b minus a end exponent plus x to the power of negative c minus a end exponent plus 6
D equals left parenthesis x to the power of b plus x to the power of negative c end exponent plus 1 right parenthesis left parenthesis x to the power of c plus x to the power of negative a end exponent plus 1 right parenthesis left parenthesis x to the power of a plus x to the power of negative b end exponent plus 1 right parenthesis
equals left parenthesis x to the power of b plus c end exponent plus x to the power of b minus a end exponent plus x to the power of b plus x to the power of negative c plus c end exponent plus x to the power of negative c minus a end exponent plus x to the power of negative c end exponent plus x to the power of c plus x to the power of negative a end exponent plus 1 right parenthesis left parenthesis x to the power of a plus x to the power of negative b end exponent plus 1 right parenthesis
equals left parenthesis x to the power of b plus c end exponent plus x to the power of b minus a end exponent plus x to the power of b plus x to the power of negative c minus a end exponent plus x to the power of negative c end exponent plus x to the power of c plus x to the power of negative a end exponent plus 2 right parenthesis left parenthesis x to the power of a plus x to the power of negative b end exponent plus 1 right parenthesis
equals x to the power of a plus b plus c end exponent plus x to the power of b plus c minus b end exponent plus x to the power of b plus c end exponent plus x to the power of b minus a plus a end exponent plus x to the power of b minus a minus b end exponent plus x to the power of b minus a end exponent plus x to the power of a plus b end exponent plus x to the power of b minus b end exponent plus x to the power of b
plus x to the power of negative c minus a plus a end exponent plus x to the power of negative c minus a minus b end exponent plus x to the power of negative c minus a end exponent plus x to the power of negative c plus a end exponent plus x to the power of negative c minus b end exponent plus x to the power of negative c end exponent plus x to the power of c plus a end exponent plus x to the power of c minus b end exponent plus x to the power of c
plus x to the power of negative a plus a end exponent plus x to the power of negative a minus b end exponent plus x to the power of negative a end exponent plus 2 x to the power of a plus 2 x to the power of negative b end exponent plus 2
equals x to the power of 0 plus x to the power of c plus x to the power of b plus c end exponent plus x to the power of b plus x to the power of negative a end exponent plus x to the power of b minus a end exponent plus x to the power of a plus b end exponent plus x to the power of 0 plus x to the power of b plus x to the power of negative c end exponent plus x to the power of negative left parenthesis a plus b plus c right parenthesis end exponent plus x to the power of negative c minus a end exponent plus x to the power of negative c plus a end exponent
plus x to the power of negative c minus b end exponent plus x to the power of negative c end exponent plus x to the power of a plus c end exponent plus x to the power of c minus b end exponent plus x to the power of c plus x to the power of 0 plus x to the power of negative a minus b end exponent plus x to the power of negative a end exponent plus 2 x to the power of a plus 2 x to the power of negative b end exponent plus 2
equals 1 plus 2 x to the power of c plus x to the power of b plus c end exponent plus 2 x to the power of b plus 2 x to the power of negative a end exponent plus x to the power of b minus a end exponent plus to the power of a plus b end exponent plus 1 plus 2 x to the power of negative c end exponent plus 1 plus x to the power of negative c minus a end exponent plus x to the power of negative c plus a end exponent plus x to the power of negative c minus b end exponent
plus x to the</h3>
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Answered by Rashmi Khot | 22nd Aug, 2016, 11:06: AM