1. $sqrt{16 - 2sqrt{20} - 2sqrt{28} + 2sqrt{35}}$
解答:
仿例题9可得: $$sqrt{16 - 2sqrt{20} - 2sqrt{28} + 2sqrt{35}} = sqrt{(sqrt7 + sqrt5 - 2)^2} = sqrt7 + sqrt5 - 2.$$
2. $sqrt{12 + sqrt{24} + sqrt{39} + sqrt{104}} - sqrt{12 - sqrt{24} + sqrt{39} - sqrt{104}}$
解答:
仿例题4可得: $$egin{cases}x^2 + y^2 = 24 + 2sqrt{39}\ xy = sqrt{55+8sqrt{39}} = sqrt{39} + 4end{cases}$$ $$Rightarrow (x - y)^2 = 16 Rightarrow x - y = 4.$$
3. $sqrt{2a - sqrt{3a^2 - 2ab - b^2}}$ ($a > b > 0$)
解答:
仿例题5可得: $$sqrt{2a - sqrt{3a^2 - 2ab - b^2}} = sqrt{2a - sqrt{(3a + b)(a - b)}} = sqrt{frac{1}{2}left(4a - 2sqrt{(3a+b)(a-b)} ight)}$$ $$= frac{sqrt2}{2}left(sqrt{3a+b} - sqrt{a - b} ight).$$
4. $sqrt{x + 2 + 3sqrt{2x-5}} - sqrt{x - 2 + sqrt{2x - 5}}$ $left(x ge dfrac{5}{2} ight)$
解答: $$sqrt{x + 2 + 3sqrt{2x-5}} - sqrt{x - 2 + sqrt{2x - 5}}$$ $$= sqrt{frac{1}{2}left(2x + 4 + 6sqrt{2x - 5} ight)} - sqrt{frac{1}{2}left(2x - 4 + 2sqrt{2x - 5} ight)}$$ $$= sqrt{frac{1}{2}left(sqrt{2x-5} + 3 ight)^2} - sqrt{frac{1}{2}left(sqrt{2x - 5} + 1 ight)^2}$$ $$= frac{sqrt2}{2}left(sqrt{2x-5} + 3 ight) - frac{sqrt2}{2}left(sqrt{2x - 5} + 1 ight) = sqrt2.$$
5. $dfrac{left(11+6sqrt2 ight)sqrt{11-6sqrt2} - left(11-6sqrt2 ight)sqrt{11+6sqrt2}}{left(sqrt{sqrt5 + 2}+ sqrt{sqrt5 - 2} ight) div sqrt{sqrt5 + 1}}$
解答: $$11 pm 6sqrt2 = left(3 pm sqrt2 ight)^2,$$ $$left(sqrt{sqrt5 + 2} + sqrt{sqrt5 - 2} ight)^2 = 2sqrt5 + 2 = 2left(sqrt5 + 1 ight).$$ $$Rightarrow frac{left(3 + sqrt2 ight)^2left(3 - sqrt2 ight) - left(3 - sqrt2 ight)^2left(3 + sqrt2 ight)}{sqrt2}$$ $$= frac{(3 + sqrt2)(3 - sqrt2)(3+sqrt2 - 3 + sqrt2)}{sqrt2} = 14.$$
6. $dfrac{2+2sqrt7 +sqrt{10}}{left(sqrt7 + sqrt{10} ight)left(sqrt7 + 2 ight)} + dfrac{4+2sqrt{13} + sqrt{10}}{left(sqrt{10} + sqrt{13} ight)left(sqrt{13} + 4 ight)}$
解答: $$frac{1}{sqrt7 + sqrt{10}} + frac{1}{sqrt{7} + 2} + frac{1}{sqrt{10} + sqrt{13}} + frac{1}{sqrt{13} + 4}$$ $$= frac{1}{3}left(sqrt{10} - sqrt7 + sqrt7 - 2 + sqrt{13} - sqrt{10} + 4 - sqrt{13} ight) = frac{2}{3}.$$
7. $dfrac{9 + 2sqrt6 + 2sqrt{14} + sqrt{21}}{2sqrt2 + sqrt3 + sqrt7}$
解答: $$left(2sqrt2 + sqrt3 + sqrt7 ight)^2 = 18 + 4sqrt6 + 4sqrt{14} + 2sqrt{21}$$ $$= 2left(9 + 2sqrt6+ 2sqrt{14} + 2sqrt{21} ight).$$ $$Rightarrow frac{dfrac{1}{2}left(2sqrt2 + sqrt3 + sqrt7 ight)^2}{2sqrt2 + sqrt3 + sqrt7}= frac{1}{2}left(2sqrt2 + sqrt3 + sqrt7 ight).$$
主讲教师:
赵胤, 理学硕士(数学) & 教育硕士(数学), 中国数学奥林匹克一级教练员, 高级中学数学教师资格.