Упр.2.156 ГДЗ Дорофеев Суворова 8 класс (Алгебра)
а) 1/(2+v3);
б) v3/(v3-v2);
в) (7-v5)/(3+v5);
г) (v11+v5)/(v11-v5);
д) (1+v3)/(v10-3);
е) (3v3-v2)/(v2+3v3).
$$\frac{1}{2+\sqrt{3}}=\frac{1}{2+\sqrt{3}}\cdot\frac{2-\sqrt{3}}{2-\sqrt{3}}=\frac{2-\sqrt{3}}{4-3}=2-\sqrt{3}.$$
$$\frac{\sqrt{3}}{\sqrt{3}-\sqrt{2}}=\frac{\sqrt{3}}{\sqrt{3}-\sqrt{2}}\cdot\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}=\frac{\sqrt{3}(\sqrt{3}+\sqrt{2})}{3-2}=3+\sqrt{6}.$$
$$\frac{7-\sqrt{5}}{3+\sqrt{5}}=\frac{(7-\sqrt{5})(3-\sqrt{5})}{(3+\sqrt{5})(3-\sqrt{5})}=\frac{21-7\sqrt{5}-3\sqrt{5}+5}{9-5}$$
$$=\frac{26-10\sqrt{5}}{4}=\frac{13-5\sqrt{5}}{2}.$$
$$\frac{\sqrt{11}+\sqrt{5}}{\sqrt{11}-\sqrt{5}}=\frac{(\sqrt{11}+\sqrt{5})^2}{(\sqrt{11}-\sqrt{5})(\sqrt{11}+\sqrt{5})}=\frac{11+2\sqrt{55}+5}{11-5}$$
$$=\frac{16+2\sqrt{55}}{6}=\frac{8+\sqrt{55}}{3}.$$
$$\frac{1+\sqrt{3}}{\sqrt{10}-3}=\frac{(1+\sqrt{3})(\sqrt{10}+3)}{(\sqrt{10}-3)(\sqrt{10}+3)}=\frac{\sqrt{10}+3+\sqrt{30}+3\sqrt{3}}{10-9}$$
$$=\sqrt{30}+\sqrt{10}+3\sqrt{3}+3.$$
$$\frac{3\sqrt{3}-\sqrt{2}}{\sqrt{2}+3\sqrt{3}}=\frac{(3\sqrt{3}-\sqrt{2})^2}{(\sqrt{2}+3\sqrt{3})(3\sqrt{3}-\sqrt{2})}=\frac{27-6\sqrt{6}+2}{9\cdot 3-2}$$
$$=\frac{29-6\sqrt{6}}{25}.$$
Ответ
а) $$2-\sqrt{3}$$; б) $$3+\sqrt{6}$$; в) $$\frac{13-5\sqrt{5}}{2}$$; г) $$\frac{8+\sqrt{55}}{3}$$; д) $$\sqrt{30}+\sqrt{10}+3\sqrt{3}+3$$; е) $$\frac{29-6\sqrt{6}}{25}$$.
