[tex]a) \\ \\ ( \sqrt{3} - \sqrt{5}) ^{2} = \\ \\ { \sqrt{3} }^{2} - 2 \sqrt{3} \sqrt{5} + { \sqrt{5} }^{2} = \\ \\ 3 - 2 \sqrt{3} \sqrt{5} + 5 = \\ \\ 3 - 2 \sqrt{3 \times 5} + 5 = \\ \\ 3 - 2 \sqrt{15} + 5 = \\ \\ 8 - 2 \sqrt{15}[/tex]
[tex]b) \\ \\ (2 \sqrt{7} + 1) ^{2} = \\ \\ (2 \sqrt{7}) ^{2} + 2 \times 2 \sqrt{7} + 1 = \\ \\ {2}^{2} { \sqrt{7} }^{2} + 2 \times 2 \sqrt{7} + 1 = \\ \\ 4 { \sqrt{7} }^{2} + 2 \times 2 \sqrt{7} + 1 = \\ \\ 4 \times 7 + 2 \times 2 \sqrt{7} + 1 = \\ \\ 28 + 4 \sqrt{7} + 1 = \\ \\ 29 + 4 \sqrt{7} [/tex]
[tex]c) \\ \\ (3 - \sqrt{7}) ^{2} = \\ \\ {3}^{2} - 2 \times 3 \sqrt{7} + { \sqrt{7} }^{2} = \\ \\ 9 - 2 \times 3 \sqrt{7} + { \sqrt{7} }^{2} = \\ \\ 9 - 2 \times 3 \sqrt{7} + 7 = \\ \\ 9 - 6 \sqrt{7} + 7 = \\ \\ 16 - 6 \sqrt{7} [/tex]
[tex]d) \\ \\ (3 \sqrt{2} + 2 \sqrt{5}) ^{2} = \\ \\ (3 \sqrt{2}) ^{2} + 2 \times 3 \sqrt{2} \times 2 \sqrt{5} + (2 \sqrt{5}) ^{2} = \\ \\ {3}^{2} { \sqrt{2} }^{2} + 2 \times 3 \sqrt{2} \times 2 \sqrt{5} + (2 \sqrt{5}) ^{2} = \\ \\ 9 { \sqrt{2} }^{2} + 2 \times 3 \sqrt{2} \times 2 \sqrt{5} + (2 \sqrt{5}) ^{2} = \\ \\ 9 \times 2 + 2 \times 3 \sqrt{2} \times 2 \sqrt{5} + (2 \sqrt{5}) ^{2} = \\ \\ 9 \times 2 + 2 \times 3 \sqrt{2} \times 2 \sqrt{5} + {2}^{2} { \sqrt{5} }^{2} = \\ \\ 9 \times 2 + 2 \times 3 \sqrt{2} \times 2 \sqrt{5} + 4 { \sqrt{5} }^{2} = \\ \\ 9 \times 2 + 2 \times 3 \sqrt{2} \times 2 \sqrt{5} + 4 \times 5 = \\ \\ 18 + 2 \times 3 \sqrt{2} \times 2 \sqrt{5} + 4 \times 5 = \\ \\ 18 + 2 \times 3 \times 2 \sqrt{2 \times 5} + 4 \times 5 = \\ \\ 18 + 2 \times 3 \times 2 \sqrt{10} + 4 \times 5 = \\ \\ 18 + 12 \sqrt{10} + 20 = \\ \\ 38 + 12 \sqrt{10} [/tex]
