a) 

$$\sqrt{1 + 2\sqrt{4 + \sqrt{9 - \sqrt{32}}}}$$ $$= \sqrt{1 + 2\sqrt{4 + \sqrt{9 - 4\sqrt{2}}}} = \sqrt{1 + 2\sqrt{4 + \sqrt{(2\sqrt{2} - 1)^2}}}$$ $$= \sqrt{1 + 2\sqrt{4 + 2\sqrt{2} - 1}$$ $$= \sqrt{1 + 2\sqrt{3 + 2\sqrt{2}}}$$ $$= \sqrt{1 + 2\sqrt{(\sqrt{2} + 1)^2}}$$ $$= \sqrt{1 + 2(\sqrt{2} + 1)} = \sqrt{3 + 2\sqrt{2}} = \sqrt{(\sqrt{2} + 1)^2} = \sqrt{2} + 1$$ 

b) 

$$\dfrac{\sqrt{3 - 2\sqrt{2}}}{\sqrt{17 - 12\sqrt{2}}} - \dfrac{\sqrt{3 + 2\sqrt{2}}}{\sqrt{17 + 12\sqrt{2}}}$$ $$= \dfrac{\sqrt{(\sqrt{2} - 1)^2}}{\sqrt{(3 - 2\sqrt{2})^2}} - \dfrac{\sqrt{(\sqrt{2} + 1)^2}}{\sqrt{(3 + 2\sqrt{2})^2}}$$ $$= \dfrac{\sqrt{2} - 1}{3 - 2\sqrt{2}} - \dfrac{\sqrt{2} + 1}{3 + 2\sqrt{2}}$$ $$= \dfrac{(\sqrt{2} - 1)(3 + 2\sqrt{2}) - (\sqrt{2} + 1)(3 - 2\sqrt{2})}{(3 - 2\sqrt{2})(3 + 2\sqrt{2})}$$ $$= \dfrac{3\sqrt{2} - 3 + 4 - 2\sqrt{2} - (3\sqrt{2} + 3 - 4 - 2\sqrt{2})}{9 - 8} = \dfrac{\sqrt{2} + 1 - \sqrt{2} + 1}{1} = 2$$ 

c) 

$$(5 + \sqrt{21}).(\sqrt{14} - \sqrt{6}) . \sqrt{5 - \sqrt{21}} = (5 + \sqrt{21} . \sqrt{2}(\sqrt{7} - \sqrt{3})\sqrt{5 - \sqrt{21}} $$$$= (5 + \sqrt{21}) (\sqrt{7} - \sqrt{3})\sqrt{10 - 2\sqrt{21}}$$ $$= (5 + \sqrt{21})(\sqrt{7} - \sqrt{3})\sqrt{(\sqrt{7} - \sqrt{3})^2} = (5 + \sqrt{21})(\sqrt{7} - \sqrt{3})(\sqrt{7} - \sqrt{3}) = (5 + \sqrt{21})(10 - 2\sqrt{21})$$ $$= (5 + \sqrt{21}).2(5 - \sqrt{21}) = 2(25 - 21) = 2.4 = 8$$