Rewrite \(27\) as \({3}^{3}\).
\[\frac{{({3}^{3})}^{\frac{5}{6}}\times {64}^{\frac{5}{6}}\times {2}^{-5}}{{(\sqrt{6}\times 4)}^{\frac{2}{3}}}\]
Use this rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\frac{{3}^{\frac{3\times 5}{6}}\times {64}^{\frac{5}{6}}\times {2}^{-5}}{{(\sqrt{6}\times 4)}^{\frac{2}{3}}}\]
Simplify \(3\times 5\) to \(15\).
\[\frac{{3}^{\frac{15}{6}}\times {64}^{\frac{5}{6}}\times {2}^{-5}}{{(\sqrt{6}\times 4)}^{\frac{2}{3}}}\]
Simplify \(\frac{15}{6}\) to \(\frac{5}{2}\).
\[\frac{{3}^{\frac{5}{2}}\times {64}^{\frac{5}{6}}\times {2}^{-5}}{{(\sqrt{6}\times 4)}^{\frac{2}{3}}}\]
Rewrite \(64\) as \({2}^{6}\).
\[\frac{{3}^{\frac{5}{2}}{({2}^{6})}^{\frac{5}{6}}\times {2}^{-5}}{{(\sqrt{6}\times 4)}^{\frac{2}{3}}}\]
Use this rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\frac{{3}^{\frac{5}{2}}\times {2}^{\frac{6\times 5}{6}}\times {2}^{-5}}{{(\sqrt{6}\times 4)}^{\frac{2}{3}}}\]
Simplify \(6\times 5\) to \(30\).
\[\frac{{3}^{\frac{5}{2}}\times {2}^{\frac{30}{6}}\times {2}^{-5}}{{(\sqrt{6}\times 4)}^{\frac{2}{3}}}\]
Simplify \(\frac{30}{6}\) to \(5\).
\[\frac{{3}^{\frac{5}{2}}\times {2}^{5}\times {2}^{-5}}{{(\sqrt{6}\times 4)}^{\frac{2}{3}}}\]
Simplify \({2}^{5}\) to \(32\).
\[\frac{{3}^{\frac{5}{2}}\times 32\times {2}^{-5}}{{(\sqrt{6}\times 4)}^{\frac{2}{3}}}\]
Use Negative Power Rule: \({x}^{-a}=\frac{1}{{x}^{a}}\).
\[\frac{{3}^{\frac{5}{2}}\times 32\times \frac{1}{{2}^{5}}}{{(\sqrt{6}\times 4)}^{\frac{2}{3}}}\]
Simplify \({2}^{5}\) to \(32\).
\[\frac{{3}^{\frac{5}{2}}\times 32\times \frac{1}{32}}{{(\sqrt{6}\times 4)}^{\frac{2}{3}}}\]
Cancel \(32\).
\[\frac{{3}^{\frac{5}{2}}}{{(\sqrt{6}\times 4)}^{\frac{2}{3}}}\]
Regroup terms.
\[\frac{{3}^{\frac{5}{2}}}{{(4\sqrt{6})}^{\frac{2}{3}}}\]
Use Multiplication Distributive Property: \({(xy)}^{a}={x}^{a}{y}^{a}\).
\[\frac{{3}^{\frac{5}{2}}}{{4}^{\frac{2}{3}}{\sqrt{6}}^{\frac{2}{3}}}\]
Rewrite \(4\) as \({2}^{2}\).
\[\frac{{3}^{\frac{5}{2}}}{{({2}^{2})}^{\frac{2}{3}}{\sqrt{6}}^{\frac{2}{3}}}\]
Use this rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\frac{{3}^{\frac{5}{2}}}{{2}^{\frac{2\times 2}{3}}{\sqrt{6}}^{\frac{2}{3}}}\]
Simplify \(2\times 2\) to \(4\).
\[\frac{{3}^{\frac{5}{2}}}{{2}^{\frac{4}{3}}{\sqrt{6}}^{\frac{2}{3}}}\]
Use this rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\frac{{3}^{\frac{5}{2}}}{{2}^{\frac{4}{3}}\times {6}^{\frac{1\times 2}{2\times 3}}}\]
Simplify \(1\times 2\) to \(2\).
\[\frac{{3}^{\frac{5}{2}}}{{2}^{\frac{4}{3}}\times {6}^{\frac{2}{2\times 3}}}\]
Simplify \(2\times 3\) to \(6\).
\[\frac{{3}^{\frac{5}{2}}}{{2}^{\frac{4}{3}}\times {6}^{\frac{2}{6}}}\]
Simplify \(\frac{2}{6}\) to \(\frac{1}{3}\).
\[\frac{{3}^{\frac{5}{2}}}{{2}^{\frac{4}{3}}\sqrt[3]{6}}\]
Decimal Form: 3.404443
3^(5/2)/(2^(4/3)*6^(1/3))
