# If 5^(10x)=4900 and 2^(sqrty)=25, what is the value of ((5^((x-1)))^5)/4^(-sqrty)?

Mar 4, 2017

${\left({5}^{x - 1}\right)}^{5} / {4}^{- \sqrt{y}} = 14$

#### Explanation:

Given that:

${5}^{10 x} = 4900$

${2}^{\sqrt{y}} = 25$

Then:

${\left({5}^{x - 1}\right)}^{5} = {5}^{5 \left(x - 1\right)}$

$\textcolor{w h i t e}{{\left({5}^{x - 1}\right)}^{5}} = {5}^{5 x - 5}$

$\textcolor{w h i t e}{{\left({5}^{x - 1}\right)}^{5}} = {5}^{5 x} \cdot {5}^{- 5}$

$\textcolor{w h i t e}{{\left({5}^{x - 1}\right)}^{5}} = {5}^{\frac{1}{2} \left(10 x\right)} \cdot {5}^{- 5}$

$\textcolor{w h i t e}{{\left({5}^{x - 1}\right)}^{5}} = {\left({5}^{10 x}\right)}^{\frac{1}{2}} \cdot {5}^{- 5}$

$\textcolor{w h i t e}{{\left({5}^{x - 1}\right)}^{5}} = {4900}^{\frac{1}{2}} \cdot {5}^{- 5}$

$\textcolor{w h i t e}{{\left({5}^{x - 1}\right)}^{5}} = 70 \cdot {5}^{- 5}$

$\textcolor{w h i t e}{{\left({5}^{x - 1}\right)}^{5}} = 14 \cdot {5}^{- 4}$

${4}^{- \sqrt{y}} = {\left({2}^{2}\right)}^{- \sqrt{y}}$

$\textcolor{w h i t e}{{4}^{- \sqrt{y}}} = {2}^{- 2 \sqrt{y}}$

$\textcolor{w h i t e}{{4}^{- \sqrt{y}}} = {\left({2}^{\sqrt{y}}\right)}^{- 2}$

$\textcolor{w h i t e}{{4}^{- \sqrt{y}}} = {25}^{- 2}$

$\textcolor{w h i t e}{{4}^{- \sqrt{y}}} = {5}^{- 4}$

So:

${\left({5}^{x - 1}\right)}^{5} / {4}^{- \sqrt{y}} = \frac{14 \cdot \textcolor{red}{\cancel{\textcolor{b l a c k}{{5}^{- 4}}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{{5}^{- 4}}}}} = 14$