Let's simplify each expression
1.
[tex]\frac{10\pi\sqrt[]{2}-8\pi\sqrt[]{2}}{2\sqrt[]{2}}=\frac{(10-8)\pi\sqrt[]{2}}{2\sqrt[]{2}}=\frac{2\pi}{2}=\pi[/tex]
2.
[tex]\frac{\sqrt[]{24}-\sqrt[]{54}}{\sqrt[]{6}}=\frac{\sqrt[]{6\cdot4}-\sqrt[]{6\cdot9}}{\sqrt[]{6}}=\frac{2\sqrt[]{6}-3\sqrt[]{6}}{\sqrt[]{6}}=\frac{(2-3)\sqrt[]{6}}{\sqrt[]{6}}=-1[/tex]
3.
[tex]\pi\sqrt[]{\frac{3}{5}}\cdot\pi\sqrt[]{\frac{5}{3}}=\pi^2\sqrt[]{\frac{3}{5}\cdot\frac{5}{3}}=\pi^2\sqrt[]{1}=\pi^2[/tex]
Using the results above, we can say that the right order is
[tex]\frac{\sqrt[]{24}-\sqrt[]{54}}{\sqrt[]{6}}<\frac{10\pi\sqrt[]{2}-8\pi\sqrt[]{2}}{2\sqrt[]{2}}<\pi\sqrt[]{\frac{3}{5}}\cdot\pi\sqrt[]{\frac{5}{3}}[/tex]
