[tex]{3}^{-\frac34}\cdot\sqrt{ {3}^{3} }\cdot\sqrt[4]{9} = {3}^{-\frac34}\cdot(3^3)^{\frac13} }\cdot9^\frac14= \big3^{-\frac34}\cdot \big3^{3\cdot\frac13}} }\cdot\left(3^2\right)^\frac14 = \big3^{-\frac34}\cdot \big3^1 }\cdot3^{2\cdot^\frac14} =\\\\= \big3^{-\frac34+1+\frac24}= \big3^{\frac34}[/tex]
[tex]\sqrt[6]{ {4}^{3}}\cdot {2}^{ \frac{3}{2} }:{8}^{- \frac {2}{3} } = \left({4}^{3}\right)^\frac16 \cdot\big2^{ \frac{3}{2} }:\left(2^3\right)^{- \frac {2}{3} } = \left(2^2\right)^{3\cdot\frac16} \cdot\big2^{ \frac{3}{2} }:\big2^{3\cdot(-\frac23) } =\\\\= \big2^{2\cdot\frac12} \cdot\big2^{ \frac{3}{2} }:\big2^{-2} = \big2^{1 +\frac{3}{2}-(-2)} = \big2^{4\frac12}[/tex]
[tex]\dfrac{ \sqrt[3]{625}\cdot {25}^{ \frac{3}{4} }\cdot\sqrt[6]{5} }{ {125}^{ \frac{2}{3}} \cdot {5}^{ - 3} }= \dfrac{\left(5^4\right)^\frac13\cdot \left(5^2\right)^\frac34\cdot\big5^\frac16} {\left(5^3\right)^{ \frac{2}{3}} \cdot \big5^{ - 3} }= \dfrac{\big5^{4\cdot\frac13}\cdot \big5^{2\cdot\frac34}\cdot\big5^\frac16} {\big5^{3\cdot\frac23} \cdot \big5^{-3} }= \dfrac{\big5^{\frac43+\frac32+\frac16}} {\big5^{2+(-3)} }=[/tex]
[tex]= \dfrac{\big5^{\frac86+\frac96+\frac16}} {\big5^{-1} }=\dfrac{\big5^{\frac{18}6}} {\big5^{-1} }=\dfrac{\big5^3}{\big5^{-1}}=\big5^{3-(-1)}=\big5^4[/tex]
