Simplify \(\sqrt{24}\) to \(2\sqrt{6}\).
\[\frac{{\sqrt[(5+2\sqrt{6})}^{2]{3}}}{\sqrt[5-2\sqrt{6}]{3}}-6-2\sqrt{6}\]
Rationalize the denominator: \(\frac{1}{5-2\sqrt{6}} \cdot \frac{5+2\sqrt{6}}{5+2\sqrt{6}}=\frac{5+2\sqrt{6}}{{5}^{2}-{(2\sqrt{6})}^{2}}\).
\[\frac{{\sqrt[(5+2\sqrt{6})}^{2]{3}}}{{3}^{\frac{5+2\sqrt{6}}{{5}^{2}-{(2\sqrt{6})}^{2}}}}-6-2\sqrt{6}\]
Simplify \({5}^{2}\) to \(25\).
\[\frac{{\sqrt[(5+2\sqrt{6})}^{2]{3}}}{{3}^{\frac{5+2\sqrt{6}}{25-{(2\sqrt{6})}^{2}}}}-6-2\sqrt{6}\]
Rationalize the denominator: \(\frac{5+2\sqrt{6}}{25-{(2\sqrt{6})}^{2}} \cdot \frac{25+{(2\sqrt{6})}^{2}}{25+{(2\sqrt{6})}^{2}}=\frac{125+120+50\sqrt{6}+48\sqrt{6}}{{25}^{2}-{({(2\sqrt{6})}^{2})}^{2}}\).
\[\frac{{\sqrt[(5+2\sqrt{6})}^{2]{3}}}{{3}^{\frac{125+120+50\sqrt{6}+48\sqrt{6}}{{25}^{2}-{({(2\sqrt{6})}^{2})}^{2}}}}-6-2\sqrt{6}\]
Collect like terms.
\[\frac{{\sqrt[(5+2\sqrt{6})}^{2]{3}}}{{3}^{\frac{(125+120)+(50\sqrt{6}+48\sqrt{6})}{{25}^{2}-{({(2\sqrt{6})}^{2})}^{2}}}}-6-2\sqrt{6}\]
Simplify \((125+120)+(50\sqrt{6}+48\sqrt{6})\) to \(245+98\sqrt{6}\).
\[\frac{{\sqrt[(5+2\sqrt{6})}^{2]{3}}}{{3}^{\frac{245+98\sqrt{6}}{{25}^{2}-{({(2\sqrt{6})}^{2})}^{2}}}}-6-2\sqrt{6}\]
Factor out the common term \(49\).
\[\frac{{\sqrt[(5+2\sqrt{6})}^{2]{3}}}{{3}^{\frac{49(5+2\sqrt{6})}{{25}^{2}-{({(2\sqrt{6})}^{2})}^{2}}}}-6-2\sqrt{6}\]
Simplify \({25}^{2}\) to \(625\).
\[\frac{{\sqrt[(5+2\sqrt{6})}^{2]{3}}}{{3}^{\frac{49(5+2\sqrt{6})}{625-{({(2\sqrt{6})}^{2})}^{2}}}}-6-2\sqrt{6}\]
Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\frac{{\sqrt[(5+2\sqrt{6})}^{2]{3}}}{{3}^{\frac{49(5+2\sqrt{6})}{625-{(2\sqrt{6})}^{4}}}}-6-2\sqrt{6}\]
Rewrite \(625-{(2\sqrt{6})}^{4}\) in the form \({a}^{2}-{b}^{2}\), where \(a=25\) and \(b=24\).
\[\frac{{\sqrt[(5+2\sqrt{6})}^{2]{3}}}{{3}^{\frac{49(5+2\sqrt{6})}{{25}^{2}-{24}^{2}}}}-6-2\sqrt{6}\]
Simplify \({25}^{2}\) to \(625\).
\[\frac{{\sqrt[(5+2\sqrt{6})}^{2]{3}}}{{3}^{\frac{49(5+2\sqrt{6})}{625-{24}^{2}}}}-6-2\sqrt{6}\]
Simplify \({24}^{2}\) to \(576\).
\[\frac{{\sqrt[(5+2\sqrt{6})}^{2]{3}}}{{3}^{\frac{49(5+2\sqrt{6})}{625-576}}}-6-2\sqrt{6}\]
Simplify \(625-576\) to \(49\).
\[\frac{{\sqrt[(5+2\sqrt{6})}^{2]{3}}}{{3}^{\frac{49(5+2\sqrt{6})}{49}}}-6-2\sqrt{6}\]
Cancel \(49\).
\[\frac{{\sqrt[(5+2\sqrt{6})}^{2]{3}}}{{3}^{5+2\sqrt{6}}}-6-2\sqrt{6}\]
Simplify \(\frac{{\sqrt[(5+2\sqrt{6})}^{2]{3}}}{{3}^{5+2\sqrt{6}}}\) to \({3}^{\frac{1}{{(5+2\sqrt{6})}^{2}}-5-2\sqrt{6}}\).
\[{3}^{\frac{1}{{(5+2\sqrt{6})}^{2}}-5-2\sqrt{6}}-6-2\sqrt{6}\]
Decimal Form: -10.898960
3^(1/(5+2*sqrt(6))^2-5-2*sqrt(6))-6-2*sqrt(6)
