If-b-x-b-k-y-k-m-and-b-1-show-that-1-x-1-y-1-m-

Question Number 56335 by Tawa1 last updated on 14/Mar/19

If b^{x} = {(\frac{b}{k})}^{y} = k^{m} and b \neq 1

show that : \frac{1}{x} = \frac{1}{y} = \frac{1}{m}

Answered by tanmay.chaudhury50@gmail.com last updated on 14/Mar/19

b^{x} = {(\frac{b}{k})}^{y} = k^{m} = r (a s s u m e d)

b^{x} = r \to b = {(r)}^{\frac{1}{x}}

k^{m} = r \to k = {(r)}^{\frac{1}{m}}

{(\frac{b}{k})}^{y} = r

{(\frac{r^{\frac{1}{x}}}{r^{\frac{1}{m}}})}^{y} = r

r^{\frac{1}{x} - \frac{1}{m}} = r^{\frac{1}{y}}

\frac{1}{x} - \frac{1}{m} = \frac{1}{y}

\frac{1}{x} + \frac{1}{y} = \frac{1}{m}

Commented by Tawa1 last updated on 14/Mar/19

God bless you sir

Answered by $@ty@m last updated on 30/Mar/19

F r o m b^{x} = k^{m} w e g e t

b = k^{\frac{m}{x}} - - (1)

F r o m {(\frac{b}{k})}^{y} = k^{m} w e g e t

b = k^{\frac{m + y}{y}} - - (2)

F r o m (1) & (2),

\frac{m}{x} = \frac{m + y}{y}

\frac{m}{x} = \frac{m}{y} + 1

\frac{m}{x} = \frac{m}{y} + \frac{m}{m}

\frac{m}{x} = m (\frac{1}{y} + \frac{1}{m})

\frac{1}{x} = \frac{1}{y} + \frac{1}{m}

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