Question

\(\left(5+4\sqrt{2}\right)\left(3+2\sqrt{1+\sqrt{2}}\right)\left(3-2\sqrt{1+\sqrt{2}}\right)\)

Answer 1

\(\left(5+4\sqrt{2}\right)\left(3+2\sqrt{1+\sqrt{2}}\right)\left(3-2\sqrt{1+\sqrt{2}}\right)=\left(5+4\sqrt{2}\right)\left[3^2-\left(2\sqrt{1+\sqrt{2}}\right)^2\right]=\left(5+4\sqrt{2}\right)\left[9-4\left(1+\sqrt{2}\right)\right]=\left(5+4\sqrt{2}\right)\left(9-4-4\sqrt{2}\right)=\left(5+4\sqrt{2}\right)\left(5-4\sqrt{2}\right)=5^2-\left(4\sqrt{2}\right)^2=25-16.2=25-32=-7\)