Question

Please anyone provide the solution for this problem fast​

Answer 1

$\large\underline{\sf{Given \:Question - }}$

If

$\sf \: \dfrac{ {25}^{n} \times {5}^{3} \times {(125)}^{3} }{5 \times {(625)}^{4} } = 125, \: find \: n.$

$\begin{gathered}\large{\sf{{\underline{Formula \: Used - }}}}  \end{gathered}$

$\boxed{ \bf{ \: {x}^{m} \times {x}^{n} = {x}^{m + n}}}$

$\boxed{ \bf{ \: {x}^{m} \div {x}^{n} = {x}^{m - n}}}$

$\boxed{ \bf{ \: {( {x}^{m} )}^{n} = {x}^{mn}}}$

$\boxed{ \bf{ \: {x}^{m} = {x}^{n}\bf\implies \:m = n}}$

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

Given expression is

$\rm :\longmapsto\:\dfrac{ {25}^{n} \times {5}^{3} \times {(125)}^{3} }{5 \times {(625)}^{4} } = 125$

can be rewritten as

$\rm :\longmapsto\:\dfrac{ {(5 \times 5)}^{n} \times {5}^{3} \times {(5 \times 5 \times 5)}^{3} }{5 \times {(5 \times 5 \times 5 \times 5)}^{4} } = 5 \times 5 \times 5$

$\rm :\longmapsto\:\dfrac{ {( {5}^{2} )}^{n} \times {5}^{3} \times {( {5}^{3} )}^{3} }{5 \times {( {5}^{4}) }^{4} } = {5}^{3}$

$\rm :\longmapsto\:\dfrac{ {5}^{2n} \times {5}^{3} \times {5}^{9} }{5 \times {5}^{16} } = {5}^{3}$

$\rm :\longmapsto\:\dfrac{ {5}^{2n + 3 + 9} }{ {5}^{1 + 16} } = {5}^{3}$

$\rm :\longmapsto\:\dfrac{ {5}^{2n + 12} }{ {5}^{17} } = {5}^{3}$

$\rm :\longmapsto\: {5}^{2n + 12 - 17} = {5}^{3}$

$\rm :\longmapsto\: {5}^{2n - 5} = {5}^{3}$

$\rm :\longmapsto\:2n - 5 = 3$

$\rm :\longmapsto\:2n = 3 + 5$

$\rm :\longmapsto\:2n = 8$

$\bf\implies \:n \: = \: 4$

Hence,

For the expression,

$\bf :\longmapsto\:\dfrac{ {25}^{n} \times {5}^{3} \times {(125)}^{3} }{5 \times {(625)}^{4} } = 125, \: value \: of\: n \: = \: 4$