Answer
$99+70\sqrt{2}$
Work Step by Step
$.................1$
$..............1....2...1$
$............1...3.....3...1.$
$..........1...4....6....4.....1$
$........1...5....10...10....5.....1$
$......1..6...15....20....15.....6.....1$
$(1+\sqrt{2})^{6}=$
$=1\cdot1^{6}(\sqrt{2})^{0}+$
$+6\cdot 1^{5}\cdot(\sqrt{2})^{1}+$
$+15\cdot 1^{4}\cdot(\sqrt{2})^{2}+$
$+20\cdot 1^{3}\cdot(\sqrt{2})^{3}+$
$+15\cdot 1^{2}\cdot(\sqrt{2})^{4}+$
$+6\cdot 1\cdot(\sqrt{2})^{5}+$
$+1\cdot(\sqrt{2})^{6}$
$=1^{6}+6\cdot 1^{5}\cdot\sqrt{2}+15\cdot 1^{4}\cdot 2+20\cdot 1^{3}\cdot 2\sqrt{2}+15\cdot 1^{2}\cdot 4+6\cdot 1\cdot 4\sqrt{2}+2^{3}$
$=1+6\sqrt{2}+30+40\sqrt{2}+60+24\sqrt{2}+8$
$=99+70\sqrt{2}$
