Math, asked by abilashap, 7 months ago

if 2 ^ x = 25 ^ y = 1000 ^ z then show that z = 2xy / (3x+6y)

Answers

Answered by MaheswariS
16

\underline{\textbf{Given:}}

\mathsf{2^x=25^y=1000^z}

\underline{\textbf{To show:}}

\mathsf{z=\dfrac{2xy}{3x+6y}}

\underline{\textbf{Solution:}}

\mathsf{Consider,}

\mathsf{2^x=25^y=1000^z=k(say)}

\implies\mathsf{2^x=k\;\;\;|\;\;\;25^y=k\;\;\;|\;\;\;1000^z=k}

\implies\mathsf{2^x=k\;\;\;|\;\;\;5^{2y}=k\;\;\;|\;\;\;1000^z=k}

\mathsf{Take\;1000^z=k}

\implies\mathsf{(5^3{\times}2^3)^z=k}

\textsf{Squaring on bothsides, we get}

\implies\mathsf{(5^3{\times}2^3)^{2z}=k^2}

\implies\mathsf{5^{6z}{\times}2^{6z}=k{\times}k}

\implies\mathsf{5^{6z}{\times}2^{6z}=2^x{\times}5^{2y}}

\textsf{Equating powers on bothsides, we get}

\mathsf{x=6z\;\;\&\;\;2y=6z}

\mathsf{x=6z\;\;\&\;\;y=3z}

\mathsf{Now,}

\mathsf{\dfrac{2xy}{3x+6y}}

\mathsf{=\dfrac{2(6z)(3z)}{3(6z)+6(3z)}}

\mathsf{=\dfrac{36\,z^2}{18\,z+18\,z}}

\mathsf{=\dfrac{36\,z^2}{36\,z}}

\mathsf{=z}

\implies\boxed{\mathsf{z=\dfrac{2xy}{3x+6y}}}

Find more:

There exist positive integers A, B and C with no common factors greater than 1, such that Alog 5 + Blog 2 = C.The sum A+B+C equals (Here the base of the log is 200)

A+B+C = 7

A+B+C = 6

A-B+C=0

A+B-C=0​

https://brainly.in/question/40036867

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