Упр.1.1 ГДЗ Мерзляк 11 класс Углубленный уровень (Алгебра)

1) 3^(v2+1)^2 : 3^(2v2);
2) ((3 7^(1/3))^(v3))^(v3);
3) (6^(v5+1)^2·36^(-v5))^(1/3);
4) ((1/2)^(v2))^(-v8).

1. $$3^{(\sqrt{2}+1)^2} : 3^{2\sqrt{2}} = 3^{(\,2+2\sqrt{2}+1\,)-2\sqrt{2}} = 3^3 = 27.$$

2. $$ \left(\left(3\sqrt[3]{7}\right)^{\sqrt{3}}\right)^{\sqrt{3}} = \left(3\sqrt[3]{7}\right)^{\sqrt{3}\cdot\sqrt{3}} = \left(3\sqrt[3]{7}\right)^3 = 3^3\cdot 7 = 189. $$

3. $$ \sqrt[3]{6^{(\sqrt{5}+1)^2}\cdot 36^{-\sqrt{5}}} = \sqrt[3]{6^{5+2\sqrt{5}+1}\cdot (6^2)^{-\sqrt{5}}} = \sqrt[3]{6^{6+2\sqrt{5}-2\sqrt{5}}} = \sqrt[3]{6^6} = 6^2 = 36. $$

4. $$ \left(\left(\frac12\right)^{\sqrt{2}}\right)^{-\sqrt{8}} = \left(\frac12\right)^{\sqrt{2}\cdot(-\sqrt{8})} = \left(\frac12\right)^{-4} = 2^4 = 16. $$

Ответ

1) $$27$$; 2) $$189$$; 3) $$36$$; 4) $$16$$.