8. Найдите значение выражения 1 (7√11)²/110; 2 48/(2√6)²; 3 (√23-4)(√23+4); 4 (√15-√7)(√15+√7); 5 (√14-3)²+6√14; 6 1/(4+√14) + 1/(4-√14); 7 1/(√37-6) - 1/(√37+6).

Решим каждое выражение по отдельности:

1. $$ \frac{(7\sqrt{11})^2}{110} = \frac{49 \cdot 11}{110} = \frac{49}{10} = 4.9 $$
2. $$ \frac{48}{(2\sqrt{6})^2} = \frac{48}{4 \cdot 6} = \frac{48}{24} = 2 $$
3. $$ (\sqrt{23}-4)(\sqrt{23}+4) = (\sqrt{23})^2 - 4^2 = 23 - 16 = 7 $$
4. $$ (\sqrt{15} - \sqrt{7})(\sqrt{15} + \sqrt{7}) = (\sqrt{15})^2 - (\sqrt{7})^2 = 15 - 7 = 8 $$
5. $$ (\sqrt{14} - 3)^2 + 6\sqrt{14} = (\sqrt{14})^2 - 2 \cdot \sqrt{14} \cdot 3 + 3^2 + 6\sqrt{14} = 14 - 6\sqrt{14} + 9 + 6\sqrt{14} = 23 $$
6. $$ \frac{1}{4+\sqrt{14}} + \frac{1}{4-\sqrt{14}} = \frac{4-\sqrt{14} + 4 + \sqrt{14}}{(4+\sqrt{14})(4-\sqrt{14})} = \frac{8}{16 - 14} = \frac{8}{2} = 4 $$
7. $$ \frac{1}{\sqrt{37}-6} - \frac{1}{\sqrt{37}+6} = \frac{\sqrt{37}+6 - (\sqrt{37}-6)}{(\sqrt{37}-6)(\sqrt{37}+6)} = \frac{\sqrt{37}+6 - \sqrt{37}+6}{37 - 36} = \frac{12}{1} = 12 $$

Ответ: 1) 4.9; 2) 2; 3) 7; 4) 8; 5) 23; 6) 4; 7) 12