a. ${{(2+\sqrt{2})}^{4}}=C_{4}^{0}{{2}^{4}}+C_{4}^{1}{{2}^{3}}.(\sqrt{2})+C_{4}^{2}{{2}^{2}}.{{(\sqrt{2})}^{2}}+C_{4}^{3}2.{{(\sqrt{2})}^{3}}+C_{4}^{4}{{(\sqrt{2})}^{4}}$
$=16+32\sqrt{2}+48+16\sqrt{2}+4$
$=68+48\sqrt{2}$.
b. ${{(2+\sqrt{2})}^{4}}$ + ${{(2-\sqrt{2})}^{4}}$
$=C_{4}^{0}{{2}^{4}}+C_{4}^{1}{{2}^{3}}.(\sqrt{2})+C_{4}^{2}{{2}^{2}}.{{(\sqrt{2})}^{2}}+C_{4}^{3}2.{{(\sqrt{2})}^{3}}+C_{4}^{4}{{(\sqrt{2})}^{4}}$
$+C_{4}^{0}{{2}^{4}}+C_{4}^{1}{{2}^{3}}.(-\sqrt{2})+C_{4}^{2}{{2}^{2}}.{{(\sqrt{2})}^{2}}+C_{4}^{3}2.{{(-\sqrt{2})}^{3}}+C_{4}^{4}{{(\sqrt{2})}^{4}}$
$=68+48\sqrt{2}$ + 0
$=16+32\sqrt{2}+48+16\sqrt{2}+4+16-32\sqrt{2}+48-16\sqrt{2}+4$
$=32+96+8$
$=136$
c. ${{(1-\sqrt{3})}^{5}}=C_{5}^{0}{{1}^{5}}+C_{5}^{1}{{1}^{4}}.(-\sqrt{3})+C_{5}^{2}{{1}^{3}}.{{(-\sqrt{3})}^{2}}$
$+C_{5}^{3}{{1}^{2}}.{{(-\sqrt{3})}^{3}}+C_{5}^{4}1.{{(-\sqrt{3})}^{4}}+C_{5}^{5}{{(-\sqrt{3})}^{5}}$
$=1-5\sqrt{3}+30-30\sqrt{3}+45-9\sqrt{3}$
$=76-44\sqrt{3}$