If 3x = 5y = (75)z , show that z = fraction numerator x y over denominator 2 x plus y end fraction .   

Asked by Trisha Gupta | 25th Feb, 2021, 10:56: AM

Expert Answer:

begin mathsize 16px style Let space 3 to the power of straight x equals 5 to the power of straight y equals 75 to the power of straight z space equals space straight k
rightwards double arrow 3 equals straight k to the power of 1 over straight x space end exponent..... left parenthesis 1 right parenthesis comma space
space space space space space 5 equals straight k to the power of 1 over straight y space end exponent space..... space left parenthesis 2 right parenthesis space and space
space space space space 75 equals straight k to the power of 1 over straight z space end exponent
rightwards double arrow 5 squared cross times 3 space equals space straight k to the power of 1 over straight z space end exponent
rightwards double arrow open parentheses straight k to the power of 1 over straight y space end exponent close parentheses squared cross times straight k to the power of 1 over straight x space end exponent equals straight k to the power of 1 over straight z space end exponent
rightwards double arrow straight k to the power of 2 over straight y plus 1 over straight x space space end exponent equals straight k to the power of 1 over straight z space end exponent space...... open parentheses straight a to the power of straight m close parentheses to the power of straight n equals space straight a to the power of mn space and space straight a to the power of straight m cross times straight a to the power of straight n equals straight a to the power of straight m plus straight n end exponent
Since space base space are space same space so space powers space will space also space be space same.
rightwards double arrow 2 over straight y plus 1 over straight x space equals 1 over straight z
rightwards double arrow fraction numerator 2 straight x plus straight y over denominator xy end fraction space equals 1 over straight z
rightwards double arrow straight z equals fraction numerator xy over denominator 2 straight x plus straight y end fraction

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Answered by Yasmeen Khan | 25th Feb, 2021, 03:14: PM