Factor out the common term \(2\).
\[\frac{2(3\sqrt{2}-2\sqrt{3}+\sqrt{5})}{9\sqrt{2}-6\sqrt{3}+3\sqrt{5}}\]
Rationalize the denominator: \(\frac{2(3\sqrt{2}-2\sqrt{3}+\sqrt{5})}{9\sqrt{2}-6\sqrt{3}+3\sqrt{5}} \cdot \frac{9\sqrt{2}-6\sqrt{3}-3\sqrt{5}}{9\sqrt{2}-6\sqrt{3}-3\sqrt{5}}=\frac{108-36\sqrt{6}-18\sqrt{10}-36\sqrt{6}+72+12\sqrt{15}+18\sqrt{10}-12\sqrt{15}-30}{162-54\sqrt{6}-27\sqrt{10}-54\sqrt{6}+108+18\sqrt{15}+27\sqrt{10}-18\sqrt{15}-45}\).
\[\frac{108-36\sqrt{6}-18\sqrt{10}-36\sqrt{6}+72+12\sqrt{15}+18\sqrt{10}-12\sqrt{15}-30}{162-54\sqrt{6}-27\sqrt{10}-54\sqrt{6}+108+18\sqrt{15}+27\sqrt{10}-18\sqrt{15}-45}\]
Factor out the common term \(6\).
\[\frac{6(18-6\sqrt{6}-3\sqrt{10}-6\sqrt{6}+12+2\sqrt{15}+3\sqrt{10}-2\sqrt{15}-5)}{162-54\sqrt{6}-27\sqrt{10}-54\sqrt{6}+108+18\sqrt{15}+27\sqrt{10}-18\sqrt{15}-45}\]
Collect like terms.
\[\frac{6((18+12-5)+(-6\sqrt{6}-6\sqrt{6})+(-3\sqrt{10}+3\sqrt{10})+(2\sqrt{15}-2\sqrt{15}))}{162-54\sqrt{6}-27\sqrt{10}-54\sqrt{6}+108+18\sqrt{15}+27\sqrt{10}-18\sqrt{15}-45}\]
Simplify \((18+12-5)+(-6\sqrt{6}-6\sqrt{6})+(-3\sqrt{10}+3\sqrt{10})+(2\sqrt{15}-2\sqrt{15})\) to \(25-12\sqrt{6}\).
\[\frac{6(25-12\sqrt{6})}{162-54\sqrt{6}-27\sqrt{10}-54\sqrt{6}+108+18\sqrt{15}+27\sqrt{10}-18\sqrt{15}-45}\]
Collect like terms.
\[\frac{6(25-12\sqrt{6})}{(162+108-45)+(-54\sqrt{6}-54\sqrt{6})+(-27\sqrt{10}+27\sqrt{10})+(18\sqrt{15}-18\sqrt{15})}\]
Simplify \((162+108-45)+(-54\sqrt{6}-54\sqrt{6})+(-27\sqrt{10}+27\sqrt{10})+(18\sqrt{15}-18\sqrt{15})\) to \(225-108\sqrt{6}\).
\[\frac{6(25-12\sqrt{6})}{225-108\sqrt{6}}\]
Rationalize the denominator: \(\frac{6(25-12\sqrt{6})}{225-108\sqrt{6}} \cdot \frac{225+108\sqrt{6}}{225+108\sqrt{6}}=\frac{33750+16200\sqrt{6}-16200\sqrt{6}-46656}{{225}^{2}-{(108\sqrt{6})}^{2}}\).
\[\frac{33750+16200\sqrt{6}-16200\sqrt{6}-46656}{{225}^{2}-{(108\sqrt{6})}^{2}}\]
Factor out the common term \(54\).
\[\frac{54(625+300\sqrt{6}-300\sqrt{6}-864)}{{225}^{2}-{(108\sqrt{6})}^{2}}\]
Collect like terms.
\[\frac{54((625-864)+(300\sqrt{6}-300\sqrt{6}))}{{225}^{2}-{(108\sqrt{6})}^{2}}\]
Simplify \((625-864)+(300\sqrt{6}-300\sqrt{6})\) to \(-239\).
\[\frac{54\times -239}{{225}^{2}-{(108\sqrt{6})}^{2}}\]
Simplify \(54\times -239\) to \(-12906\).
\[\frac{-12906}{{225}^{2}-{(108\sqrt{6})}^{2}}\]
Simplify \({225}^{2}\) to \(50625\).
\[\frac{-12906}{50625-{(108\sqrt{6})}^{2}}\]
Use Multiplication Distributive Property: \({(xy)}^{a}={x}^{a}{y}^{a}\).
\[\frac{-12906}{50625-{108}^{2}{\sqrt{6}}^{2}}\]
Simplify \({108}^{2}\) to \(11664\).
\[\frac{-12906}{50625-11664{\sqrt{6}}^{2}}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[\frac{-12906}{50625-11664\times 6}\]
Simplify \(11664\times 6\) to \(69984\).
\[\frac{-12906}{50625-69984}\]
Simplify \(50625-69984\) to \(-19359\).
\[\frac{-12906}{-19359}\]
Two negatives make a positive.
\[\frac{12906}{19359}\]
Simplify.
\[\frac{2}{3}\]
Decimal Form: 0.666667
2/3
