Алгебра 8 класс учебник Макарычев, Миндюк ответы – номер 444

а) $$ \sqrt{10+\sqrt{24}+\sqrt{40}+\sqrt{60}}= $$ = $$\sqrt{2}$$ + $$\sqrt{3}$$ + $$\sqrt{5}$$
$$ {\sqrt{10+\sqrt{24}+\sqrt{40}+\sqrt{60}^2}}^2 $$ = 10 + $$\sqrt{24}$$ + $$\sqrt{40}$$ + $$\sqrt{60}$$ = 10 + $$\sqrt{4 ⋅ 6}$$ + $$\sqrt{4 ⋅ 10}$$ + $$\sqrt{4 ⋅ 15}$$ = 10 + 2$$\sqrt{6}$$ + 2$$\sqrt{10}$$ + 2$$\sqrt{15}$$
$$\sqrt{2}$$ + $$\sqrt{3}$$ + $$\sqrt{5}$$ = ($$\sqrt{2}$$ + $$\sqrt{3}$$ + $$\sqrt{5}$$)² = ($$\sqrt{2}$$ + $$\sqrt{3}$$)² + 2($$\sqrt{2}$$ + $$\sqrt{3}$$) ⋅ $$\sqrt{5}$$ + $$\sqrt{5}$$² = $$\sqrt{2}$$² + 2$$\sqrt{2}$$ ⋅ $$\sqrt{3}$$ + $$\sqrt{3}$$² + 2$$\sqrt{10}$$ + 2$$\sqrt{15}$$ + 5 = 2 + 2$$\sqrt{6}$$ + 3 + 2$$\sqrt{10}$$ + 2$$\sqrt{15}$$ + 5 = 10 + 2$$\sqrt{6}$$ + 2$$\sqrt{10}$$ + 2$$\sqrt{15}$$
10 + 2$$\sqrt{6}$$ + 2$$\sqrt{10}$$ + 2$$\sqrt{15}$$ = 10 + 2$$\sqrt{6}$$ + 2$$\sqrt{10}$$ + 2$$\sqrt{15}$$
б) $$ \sqrt{9+\sqrt{12}-\sqrt{20}-\sqrt{60}}=\sqrt{9+\sqrt{12}-\sqrt{20}-\sqrt{60}^2} $$ = 9 + $$\sqrt{12}$$ − $$\sqrt{20}$$ − $$\sqrt{60}$$ = 9 + $$\sqrt{4 ⋅ 3}$$ − $$\sqrt{4 ⋅ 5}$$ − $$\sqrt{4 ⋅ 5}$$ = 9 + 2$$\sqrt{3}$$ − 2$$\sqrt{5}$$ − 2$$\sqrt{15}$$
1 + $$\sqrt{3}$$ − $$\sqrt{5}$$ = (1 + $$\sqrt{3}$$ − $$\sqrt{5}$$)² = (1 + $$\sqrt{3}$$)² − 2(1 + $$\sqrt{3}$$) ⋅ $$\sqrt{3}$$ + $$\sqrt{3}$$² = 1² + 2 ⋅ 1 ⋅ $$\sqrt{3}$$ + $$\sqrt{3}$$² − 2$$\sqrt{5}$$ − 2$$\sqrt{15}$$ + 5 = 1 + 2$$\sqrt{3}$$ + 3 − 2$$\sqrt{5}$$ − 2$$\sqrt{15}$$ + 5 = 9 + 2$$\sqrt{3}$$ − 2$$\sqrt{5}$$ − 2$$\sqrt{15}$$
9 + 2$$\sqrt{3}$$ − 2$$\sqrt{5}$$ − 2$$\sqrt{15}$$ = 9 + 2$$\sqrt{3}$$ − 2$$\sqrt{5}$$ − 2$$\sqrt{15}$$