Упражнение 16 Повторение ГДЗ Никольский 10 класс (Алгебра)

1. $$3\cdot \frac{\sqrt{8+2\sqrt7}}{\sqrt{8-2\sqrt7}}-\frac{\sqrt{3+\sqrt7}}{\sqrt{3-\sqrt7}}\cdot \sqrt2$$
   $$=3\cdot \frac{\sqrt{(1+\sqrt7)^2}}{\sqrt{(\sqrt7-1)^2}}-\frac{\sqrt{3+\sqrt7}\cdot \sqrt2}{\sqrt{3-\sqrt7}}$$
   $$=3\cdot \frac{1+\sqrt7}{\sqrt7-1}-\frac{\sqrt{(3+\sqrt7)\cdot 2}}{\sqrt{3-\sqrt7}}$$
   $$=\frac{3(1+\sqrt7)}{\sqrt7-1}-\frac{\sqrt{6+2\sqrt7}}{\sqrt{3-\sqrt7}}$$
   $$=\frac{3(1+\sqrt7)}{\sqrt7-1}-\frac{\sqrt{(\sqrt7+1)^2}}{\sqrt{3-\sqrt7}}$$
   $$=\frac{3(1+\sqrt7)}{\sqrt7-1}-\frac{\sqrt7+1}{\sqrt7-1}$$
   $$=\frac{3+3\sqrt7-\sqrt7-1}{\sqrt7-1}=\frac{2+2\sqrt7}{\sqrt7-1}=2\cdot \frac{\sqrt7+1}{\sqrt7-1}=1.$$

2. $$\sqrt{3-2\sqrt2}+4\cdot \frac{\sqrt{6+\sqrt2}}{\sqrt{6-\sqrt2}}\cdot \sqrt{\frac72}$$
   $$=\sqrt{(\sqrt2-1)^2}+4\cdot \frac{\sqrt{6+\sqrt2}\cdot \sqrt{7}}{\sqrt{6-\sqrt2}\cdot \sqrt2}$$
   $$=\sqrt2-1+4\cdot \frac{\sqrt{(6+\sqrt2)(6+\sqrt2)}}{\sqrt{(6-\sqrt2)(6+\sqrt2)}\cdot \sqrt2}$$
   $$=\sqrt2-1+4\cdot \frac{6+\sqrt2}{\sqrt{36-2}\cdot \sqrt2}$$
   $$=\sqrt2-1+4\cdot \frac{6+\sqrt2}{\sqrt{34}\cdot \sqrt2}=\sqrt2-1+12+\sqrt2=15.$$

3. $$(\sqrt[3]{2}-\sqrt[3]{5})(\sqrt[3]{4}+\sqrt[3]{10}+\sqrt[3]{25})$$
   $$=(\sqrt[3]{2})^3-(\sqrt[3]{5})^3=2-5=-3.$$

4. $$\sqrt[6]{4-2\sqrt3}\cdot \sqrt[3]{7+\sqrt3}\cdot \sqrt[3]{4}$$
   $$=\sqrt[6]{(\sqrt3-1)^2}\cdot \sqrt[3]{7+\sqrt3}\cdot \sqrt[3]{4}$$
   $$=\sqrt[3]{\sqrt3-1}\cdot \sqrt[3]{7+\sqrt3}\cdot \sqrt[3]{4}$$
   $$=\sqrt[3]{4(\sqrt3-1)(7+\sqrt3)}=\sqrt[3]{8}=2.$$

5. $$\sqrt{2-\sqrt{2+\sqrt3}}\cdot \sqrt{2+\sqrt{2+\sqrt3}}\cdot \sqrt{2+\sqrt3}$$
   $$=\sqrt{(2-\sqrt{2+\sqrt3})(2+\sqrt{2+\sqrt3})}\cdot \sqrt{2+\sqrt3}$$
   $$=\sqrt{4-(2+\sqrt3)}\cdot \sqrt{2+\sqrt3}$$
   $$=\sqrt{2-\sqrt3}\cdot \sqrt{2+\sqrt3}=\sqrt{4-3}=1.$$

6. $$\left(\sqrt[6]{(2+\sqrt3)^2}+2\sqrt[3]{2+\sqrt3}\right)\sqrt[3]{2-\sqrt3}$$
   $$=\left(\sqrt[3]{2+\sqrt3}+2\sqrt[3]{2+\sqrt3}\right)\sqrt[3]{2-\sqrt3}$$
   $$=3\sqrt[3]{(2+\sqrt3)(2-\sqrt3)}=3\sqrt[3]{1}=3.$$

Ответ

1) $$1$$; 2) $$15$$; 3) $$-3$$; 4) $$2$$; 5) $$1$$; 6) $$3$$.