# How do you simplify \frac { ( 9s ^ { - 5} t ^ { 4} ) ^ { 3/ 2} } { ( 27s ^ { 6} t ^ { - 2} ) ^ { 2/ 3} }?

Jan 9, 2018

Assuming $s$ and $t$ are both positive;

$\frac{{\left(9 {s}^{- 5} {t}^{4}\right)}^{\frac{3}{2}}}{{\left(27 {s}^{6} {t}^{- 2}\right)}^{\frac{2}{3}}} = \frac{3 {t}^{\frac{22}{3}}}{s} ^ \left(\frac{23}{2}\right)$

#### Explanation:

((9s^(-5)t^4)^(3/2))/((27s^6t^(-2))^(2/3))=((3^2s^(-5)t^4)^(3/2))/((3^3s^6t^(-2))^(2/3)

$\frac{{\left({3}^{2} {s}^{- 5} {t}^{4}\right)}^{\frac{3}{2}}}{{\left({3}^{3} {s}^{6} {t}^{- 2}\right)}^{\frac{2}{3}}} = \frac{{3}^{3} {s}^{- \frac{15}{2}} {t}^{6}}{{3}^{2} {s}^{4} {t}^{- \frac{4}{3}}}$

(3^3s^(-15/2)t^6)/(3^2s^4t^(-4/3))=3^(3-2)s^((-15-8)/2)t^((18-(-4))/3

${3}^{3 - 2} {s}^{\frac{- 15 - 8}{2}} {t}^{\frac{18 - \left(- 4\right)}{3}} = 3 {s}^{- \frac{23}{2}} {t}^{\frac{22}{3}}$

$3 {s}^{- \frac{23}{2}} {t}^{\frac{22}{3}} = \frac{3 {t}^{\frac{22}{3}}}{s} ^ \left(\frac{23}{2}\right)$

Hope it helps :)