$$\left(\mathrm{1}+{i}\right)^{\mathrm{80}} \:=\:\underset{{p}=\mathrm{0}} {\overset{\mathrm{80}} {\sum}}\mathrm{C}_{{p}} ^{\mathrm{80}} {i}^{{p}} \\$$$$\left(\mathrm{1}+{i}\right)^{\mathrm{80}} \:=\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{80}} {\sum}}\mathrm{C}_{\mathrm{2}{k}} ^{\mathrm{80}} {i}^{\mathrm{2}{k}} +\underset{{k}=\mathrm{0}} {\overset{\mathrm{39}} {\sum}}\mathrm{C}_{\mathrm{2}{k}+\mathrm{1}} ^{\mathrm{80}} {i}^{\mathrm{2}{k}+\mathrm{1}} \\$$$$\left(\mathrm{1}+{i}\right)^{\mathrm{80}} \:=\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{80}} {\sum}}\left(−\mathrm{1}\right)^{{k}} \mathrm{C}_{\mathrm{2}{k}} ^{\mathrm{80}} +\underset{{k}=\mathrm{0}} {\overset{\mathrm{39}} {\sum}}\left(−\mathrm{1}\right)^{{k}} \mathrm{C}_{\mathrm{2}{k}+\mathrm{1}} ^{\mathrm{80}} {i} \\$$$$\mathrm{Re}\left[\left(\mathrm{1}+{i}\right)^{\mathrm{80}} \right]\:=\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{80}} {\sum}}\left(−\mathrm{1}\right)^{{k}} \mathrm{C}_{\mathrm{2}{k}} ^{\mathrm{80}} \\$$$$\mathrm{Re}\left[\left(\sqrt{\mathrm{2}}\mathrm{e}^{{i}\frac{\pi}{\mathrm{4}}} \right)^{\mathrm{80}} \right]\:=\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{80}} {\sum}}\left(−\mathrm{1}\right)^{{k}} \mathrm{C}_{\mathrm{2}{k}} ^{\mathrm{80}} \\$$$$\mathrm{2}^{\mathrm{40}} \:=\:\underset{{k}=\mathrm{0}} {\overset{\mathrm{80}} {\sum}}\left(−\mathrm{1}\right)^{{k}} \mathrm{C}_{\mathrm{2}{k}} ^{\mathrm{80}} \\$$