If x^p = y^q = (xy)^pq , show that p+q = 1

Asked by sinhayashvi90 | 23rd Jun, 2022, 10:53: PM

Expert Answer:

Hint
straight x to the power of straight p equals straight k
straight x equals left parenthesis straight k right parenthesis to the power of 1 divided by straight p end exponent space straight y equals left parenthesis straight k right parenthesis to the power of 1 divided by straight q end exponent space space
left parenthesis xy right parenthesis to the power of pq space space equals space straight k
open parentheses left parenthesis straight k right parenthesis to the power of 1 divided by straight p end exponent cross times left parenthesis straight k right parenthesis to the power of 1 divided by straight q end exponent close parentheses to the power of pq space equals straight k
open parentheses left parenthesis straight k right parenthesis to the power of 1 over straight p plus 1 over straight q end exponent close parentheses to the power of pq equals straight k
open parentheses left parenthesis straight k right parenthesis to the power of fraction numerator straight p plus straight q over denominator pq end fraction end exponent close parentheses to the power of pq equals straight k
this space gives comma
straight p plus straight q equals 1

Answered by  | 24th Jun, 2022, 01:37: AM

Queries asked on Sunday & after 7pm from Monday to Saturday will be answered after 12pm the next working day.