(11) The first digit after the decimal point of $(\sqrt{2}+\sqrt{3})^{2010}$ is
119 Prompt: Since $(\sqrt{2}+\sqrt{3})^{2010}+(\sqrt{2}-\sqrt{3})^{2010}$ is an integer, then the fractional part of $(\sqrt{2}+\sqrt{3})^{2010}$ is $1-(\sqrt{2}-\sqrt{3})^{2010}$, and because
$$
0<(\sqrt{2}-\sqrt{3})^{2010}<0.2^{1005}<(0.008)^{300},
$$

therefore
$$
0.9<1-(\sqrt{2}-\sqrt{3})^{2010}<1 \text {, }
$$

it follows that the first digit after the decimal point of $(\sqrt{2}+\sqrt{3})^{2010}$ is 9.