[tex] \frac{1 \sqrt{2} }{2} + \frac{2 \sqrt{8} }{8} + \frac{3 \sqrt{18} }{18} + \frac{4 \sqrt{38} }{38} = \frac{ \sqrt{2} }{2} + \frac{2 \cdot 2 \sqrt{2} }{8} + \frac{3 \cdot 3 \sqrt{2} }{18} + \frac{2 \sqrt{38} }{19} = \\ = \frac{4 \sqrt{2} }{8} + \frac{9 \sqrt{2} }{18} + \frac{2 \sqrt{38} }{19} + \frac{ \sqrt
{2} }{2} = \frac{ \sqrt{2} }{2} + \frac{ \sqrt{2} }{2} + \frac{2 \sqrt{38} }{19} + \frac{ \sqrt{2} }{2} = [/tex]
[tex]= \frac{2 \sqrt{38} }{19} + \frac{ \sqrt{2} + \sqrt{2} + \sqrt{2} }{2} = \frac{2 \sqrt{38} }{19} + \frac{3 \sqrt{2} }{2} = \boxed{\frac{4 \sqrt{38} +57 \sqrt{2} }{38} }[/tex]
