[tex]a) \sqrt{1 + 3 + 5 + ... + 21} [/tex]
[tex]1 + 3 + 5 + ... + 2n - 1 = {n}^{2} [/tex]
[tex]2n - 1 = 21[/tex]
[tex]2n = 21 + 1[/tex]
[tex]2n = 22 \: | \div 2[/tex]
[tex]n = 11[/tex]
[tex] \sqrt{ {n}^{2} } = \sqrt{ {11}^{2} } = \sqrt{121} = 11 \: \in \: \mathbb{N}[/tex]
[tex]b) \sqrt[3]{27} - \sqrt{12} + 2 \sqrt{3} [/tex]
[tex] = \sqrt[3]{ {3}^{3} } - 2 \sqrt{3} + 2 \sqrt{3} [/tex]
[tex] = {3}^{ \frac{3}{3} } + 0[/tex]
[tex] = {3}^{1} [/tex]
[tex] = 3 \: \in \: \mathbb{Z}[/tex]
[tex]c) {(1 + \sqrt{2} )}^{2} + {(1 - \sqrt{2} )}^{2} [/tex]
[tex] = {1}^{2} + 2 \times 1 \times \sqrt{2} + ({ \sqrt{2} )}^{2} +{1}^{2}- 2 \times 1 \times \sqrt{2} + { (\sqrt{2}) }^{2} [/tex]
[tex] = 1 + 2 \sqrt{2} + 2 + 1 - 2 \sqrt{2} + 2[/tex]
[tex] = 1 + 1 + 2 + 2 + 2 \sqrt{2} - 2 \sqrt{2} [/tex]
[tex] = 6 + 0[/tex]
[tex] = 6 \: \in \: \mathbb{N}[/tex]
