2)
[tex] {(a + \frac{1}{a}) }^{2} = {3}^{2} \\ {a}^{2} + 2 \times a \times \frac{1}{a} + \frac{1}{ {a}^{2} } = 9 \\ {a}^{2} + 2 + \frac{1}{ {a}^{2} } = 9 \\ {a}^{2} + \frac{1 }{ {a}^{2} } = 7[/tex]
4)
[tex]a = \sqrt{4 + 2 \sqrt{3} } \\ 2 \times a \times b = 2 \sqrt{3} \\ a \times b = \sqrt{3} \\ a = \sqrt{3} \\ b = 1[/tex]
[tex] \sqrt{4 + 2 \sqrt{3} } = | \sqrt{3} + 1| = \sqrt{3} + 1[/tex]
[tex]b = \sqrt{4 - 2 \sqrt{3} } \\ 2 \times a \times b = 2 \sqrt{3} \\ a \times b = \sqrt{3} \\ a = \sqrt{3} \\ b = 1[/tex]
[tex] \sqrt{4 - 2 \sqrt{3} } = | \sqrt{3} - 1 | = \sqrt{3} - 1[/tex]
a)
[tex]{(a + b)}^{2} = {( \sqrt{3} + 1 + \sqrt{3} - 1) }^{2} = {(2 \sqrt{3}) }^{2} = 12[/tex]
b)
[tex]mg = \sqrt{ (\sqrt{3} + 1) \times ( \sqrt{3 } - 1)} \\ mg = \sqrt{ { (\sqrt{3} )}^{2} - 1} \\ mg = \sqrt{3 - 1} = \sqrt{2} [/tex]
