Question

Solve the equations

$\sf \: {10}^{ \frac{2}{x} } + {25}^{ \frac{1}{x} } = \frac{17}{4} \times {50}^{ \frac{1}{x} }$

Spammers please far away.

Brainly Mods, Brainly stars or best user or genius.

Please answer.

Don't co py and pa ste from Bro wser!!!!

Otherwise reported. ​

Answer 1

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

Given equation is

$\rm :\longmapsto\:{\bigg[10\bigg]}^{\dfrac{2}{x} } + {\bigg[25\bigg]}^{\dfrac{1}{x} } = \dfrac{17}{4} \times {\bigg[50\bigg]}^{\dfrac{1}{x} }$

can be rewritten as

$\rm :\longmapsto\:{\bigg[ {10}^{2} \bigg]}^{\dfrac{1}{x} } + {\bigg[25\bigg]}^{\dfrac{1}{x} } = \dfrac{17}{4} \times {\bigg[50\bigg]}^{\dfrac{1}{x} }$

$\rm :\longmapsto\:{\bigg[100 \bigg]}^{\dfrac{1}{x} } + {\bigg[25\bigg]}^{\dfrac{1}{x} } = \dfrac{17}{4} \times {\bigg[50\bigg]}^{\dfrac{1}{x} }$

can be further rewritten as

$\red{ \boxed{ \sf{ \:on \: dividing \: each \: term \: by \: {\bigg[50\bigg]}^{\dfrac{1}{x} }, \: we \: get}}}$

We know that,

$\red{ \boxed{ \sf{ \: \frac{ {x}^{m} }{ {y}^{m} } = {\bigg[\dfrac{x}{y} \bigg]}^{m}}}}$

So, apply these, we get

$\rm :\longmapsto\:{\bigg[2\bigg]}^{\dfrac{1}{x} } + \dfrac{1}{{\bigg[2\bigg]}^{\dfrac{1}{x} } } = \dfrac{17}{4}$

Let we assume that,

$\red{\rm :\longmapsto\:{\bigg[2\bigg]}^{\dfrac{1}{x} } = y - - - - (1)}$

So, above equation can be rewritten as

$\rm :\longmapsto\:y + \dfrac{1}{y} = \dfrac{17}{4}$

$\rm :\longmapsto\:\dfrac{ {y}^{2} + 1}{y} = \dfrac{17}{4}$

$\rm :\longmapsto\: {4y}^{2} + 4 = 17y$

$\rm :\longmapsto\: {4y}^{2} - 17y + 4 = 0$

$\rm :\longmapsto\: {4y}^{2} - 16y - y + 4 = 0$

$\rm :\longmapsto\:4y(y - 4) - 1(y - 4) = 0$

$\rm :\longmapsto\:(y - 4)(4y - 1) = 0$

$\bf\implies \:y = 4 \: \: \: \: or \: \: \: \: y = \dfrac{1}{4}$

$\bf\implies \:{\bigg[2\bigg]}^{\dfrac{1}{x} } = 4 \: \: \: \: or \: \: \: \: {\bigg[2\bigg]}^{\dfrac{1}{x} } = \dfrac{1}{4}$

$\bf\implies \:{\bigg[2\bigg]}^{\dfrac{1}{x} } = {2}^{2} \: \: \: \: or \: \: \: \: {\bigg[2\bigg]}^{\dfrac{1}{x} } = \dfrac{1}{ {2}^{2} }$

$\bf\implies \:\dfrac{1}{x} = 2 \: \: \: \: or \: \: \: \: \dfrac{1}{x} = - 2$

$\bf\implies \:x = \dfrac{1}{2} \: \: \: or \: \: \: - \: \dfrac{1}{2}$