Question

please solve this problem questions.
please help me friends.​

Answer 1

$\large\underline{\sf{Solution-}}$

It is given that

$\rm :\longmapsto\: {2}^{x} = {5}^{y} = {10}^{z}$

Let suppose that

$\rm :\longmapsto\: {2}^{x} = {5}^{y} = {10}^{z} = k$

So,

$\rm :\longmapsto\: {2}^{x} = k \: \rm \implies\:2 = {\bigg(k\bigg) }^{\dfrac{1}{x} } - - - - (i)$

$\rm :\longmapsto\: {5}^{y} = k \: \rm \implies\:5 = {\bigg(k\bigg) }^{\dfrac{1}{y} } - - - - (ii)$

$\rm :\longmapsto\: {10}^{z} = k \: \rm \implies\:10 = {\bigg(k\bigg) }^{\dfrac{1}{z} }$

$\rm :\longmapsto\: 2 \times 5 = {\bigg(k\bigg) }^{\dfrac{1}{z} }$

$\rm :\longmapsto\: {\bigg(k\bigg) }^{\dfrac{1}{x} } \times {\bigg(k\bigg) }^{\dfrac{1}{y} } = {\bigg(k\bigg) }^{\dfrac{1}{z} }$

$\rm :\longmapsto\: {\bigg(k\bigg) }^{\dfrac{1}{x} + \dfrac{1}{y} } = {\bigg(k\bigg) }^{\dfrac{1}{z} }$

${\bigg \{ \because \: {a}^{m} \times {a}^{n} = {a}^{m + n} \bigg \}}$

$\bf\implies \:\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{z}$

Proved.