1.255. Выполните умножение: a) (√6+2)(√6-1); B) (2-5√3)(4√3–7); д) (3√2 + √3)(√2 - √3); 6) (3√5-2)(√5+5); г) (6√11+5)(3-√11); e) (7√7-√5)(3√7–4√5).

Решение:

1. a) \( (\sqrt{6}+2)(\sqrt{6}-1) \)


• \( (\sqrt{6}+2)(\sqrt{6}-1) = (\sqrt{6})^2 - \sqrt{6} + 2\sqrt{6} - 2 \)
• \( = 6 + \sqrt{6} - 2 \)
• \( = 4 + \sqrt{6} \)
2. б) \( (3\sqrt{5}-2)(\sqrt{5}+5) \)


• \( (3\sqrt{5}-2)(\sqrt{5}+5) = 3(\sqrt{5})^2 + 15\sqrt{5} - 2\sqrt{5} - 10 \)
• \( = 3 \cdot 5 + 13\sqrt{5} - 10 \)
• \( = 15 + 13\sqrt{5} - 10 \)
• \( = 5 + 13\sqrt{5} \)
3. в) \( (2-5\sqrt{3})(4\sqrt{3}-7) \)


• \( (2-5\sqrt{3})(4\sqrt{3}-7) = 8\sqrt{3} - 14 - 20(\sqrt{3})^2 + 35\sqrt{3} \)
• \( = 8\sqrt{3} - 14 - 20 \cdot 3 + 35\sqrt{3} \)
• \( = 8\sqrt{3} - 14 - 60 + 35\sqrt{3} \)
• \( = 43\sqrt{3} - 74 \)
4. г) \( (6\sqrt{11}+5)(3-\sqrt{11}) \)


• \( (6\sqrt{11}+5)(3-\sqrt{11}) = 18\sqrt{11} - 6(\sqrt{11})^2 + 15 - 5\sqrt{11} \)
• \( = 18\sqrt{11} - 6 \cdot 11 + 15 - 5\sqrt{11} \)
• \( = 18\sqrt{11} - 66 + 15 - 5\sqrt{11} \)
• \( = 13\sqrt{11} - 51 \)
5. д) \( (3\sqrt{2}+\sqrt{3})(\sqrt{2}-\sqrt{3}) \)


• \( (3\sqrt{2}+\sqrt{3})(\sqrt{2}-\sqrt{3}) = 3(\sqrt{2})^2 - 3\sqrt{2}\sqrt{3} + \sqrt{3}\sqrt{2} - (\sqrt{3})^2 \)
• \( = 3 \cdot 2 - 3\sqrt{6} + \sqrt{6} - 3 \)
• \( = 6 - 2\sqrt{6} - 3 \)
• \( = 3 - 2\sqrt{6} \)
6. е) \( (7\sqrt{7}-2\sqrt{5})(3\sqrt{7}-4\sqrt{5}) \)


• \( (7\sqrt{7}-2\sqrt{5})(3\sqrt{7}-4\sqrt{5}) = 21(\sqrt{7})^2 - 28\sqrt{35} - 6\sqrt{35} + 8(\sqrt{5})^2 \)
• \( = 21 \cdot 7 - 34\sqrt{35} + 8 \cdot 5 \)
• \( = 147 - 34\sqrt{35} + 40 \)
• \( = 187 - 34\sqrt{35} \)

Ответ: а) $$4 + \sqrt{6}$$; б) $$5 + 13\sqrt{5}$$; в) $$43\sqrt{3} - 74$$; г) $$13\sqrt{11} - 51$$; д) $$3 - 2\sqrt{6}$$; е) $$187 - 34\sqrt{35}$$.