First note that
$\displaystyle 16-2\sqrt{29}+2\sqrt{55-10\sqrt{29}}=\left(\sqrt{11-2\sqrt{29}}+\sqrt{5}\right)^2$
So then
$\displaystyle \sqrt{11+2\sqrt{29}}+\sqrt{16-2\sqrt{29}+2\sqrt{55-10\sqrt{29}}}=\sqrt{11+2\sqrt{29}}+\sqrt{11-2\sqrt{29}}+\sqrt{5}$
Furthermore,
$\displaystyle \sqrt{11+2\sqrt{29}}+\sqrt{11-2\sqrt{29}}=\sqrt{\left(\sqrt{11+2\sqrt{29}}+\sqrt {11-2\sqrt{29}}\right)^2}=\sqrt{22+2\sqrt{5}}$
Finally,
$\displaystyle \sqrt{11+2\sqrt{29}}+\sqrt{11-2\sqrt{29}}+\sqrt{5}=\sqrt{5}+\sqrt{22+2\sqrt{5}}$
but actually one of my teachers once gave me a question like this he was offered on an advanced mathematics exam for college...I think it was the GRE for mathematics....but he said that he did it similar to the way the guy above me did...but this answer was also acceptable...since the left side is equal to 7.38118 and so is the left side you have proved this...my teacher said the test givers wanted you to realize that not everything is as complicated as it seems and that if you follow their instructions and actually read the question things can be done in a much simpler way...hope this helps