| Предыдущая версия справа и слеваПредыдущая версияСледующая версия | Предыдущая версия |
| math-public:uproscheniya_s_bukvami [2016/11/07 19:25] – labreslav | math-public:uproscheniya_s_bukvami [2020/01/17 23:30] (текущий) – labreslav |
|---|
| | ^ Номер ^ --------- Условие --------- ^ --------- Ответ --------- ^ |
| | | \\ 1. | \\ $\left(\dfrac{\sqrt{a}}{\sqrt{a}-\sqrt{b}}-\dfrac{\sqrt{b}}{\sqrt{a}+\sqrt{b}}\right)\cdot\dfrac{a-b}{a^2+ab}$\\ | \\ $\dfrac{1}{a}$ | |
| | | \\ 2. | \\ $\left(\dfrac{\sqrt{a}+1}{\sqrt{a}-1}-\dfrac{\sqrt{a}-1}{\sqrt{a}+1}+4\sqrt{a}\right)\left(\sqrt{\dfrac{a}{4}}-\dfrac{1}{\sqrt{4a}}\right)$\\ | \\ $2a$ | |
| | | \\ 3. | \\ $\left(\dfrac{\sqrt{a}}{\sqrt{a}+\sqrt{b}}+\dfrac{\sqrt{b}}{\sqrt{a}-\sqrt{b}}+\dfrac{2\sqrt{ab}}{a-b}\right)\left(\sqrt{a}-\dfrac{\sqrt{ab}+b}{\sqrt{a}+\sqrt{b}}\right)$\\ | \\ $\sqrt{a}+\sqrt{b}$ | |
| | | \\ 4. | \\ $\dfrac{\dfrac{2a}{\sqrt{a+b}}+\sqrt{a-b}}{1+\sqrt{\dfrac{a-b}{a+b}}}\cdot\dfrac{2b}{(a+b)\sqrt{a+b}-(a-b)\sqrt{a-b}}$\\ | \\ $1$ | |
| | | \\ 5. | \\ $\dfrac{\dfrac{a\sqrt{a}+b\sqrt{b}}{\sqrt{a}+\sqrt{b}}}{a-b}+\dfrac{2\sqrt{b}}{\sqrt{a}+\sqrt{b}}-\dfrac{\sqrt{ab}}{a-b}$\\ | \\ $1$ | |
| | | \\ 6. | \\ $\dfrac{a-b}{a+b+\sqrt{(a+b)^2-(a-b)^2}}+\dfrac{2(a\sqrt{a}-b\sqrt{b})}{(\sqrt{a}+\sqrt{b})\left(a+b-\sqrt{(a+b)^2-(a-b)^2}\right)}$\\ | \\ $\dfrac{3a+3b}{a-b}$ | |
| | | \\ 7. | \\ $\sqrt{\dfrac{3b+a^3}{2a}+\sqrt{3ab}}-\sqrt{\dfrac{3b+a^3}{2a}-\sqrt{3ab}}$ , при $0<a^3<3b$\\ | \\ $a\sqrt{2}$ | |
| | | \\ 8. | \\ $\dfrac{\sqrt{z}-2}{4z-16\sqrt{z}+16}:\left(\dfrac{\sqrt{z}}{2\sqrt{z}-4}-\dfrac{z-12}{2z-8}-\dfrac{2}{z+2\sqrt{z}}\right)$\\ | \\ $\dfrac{\sqrt{z}}{4(\sqrt{z}+2)}$ | |
| | | \\ 9. | \\ $\dfrac{(\sqrt{a}-\sqrt{b})^3+\dfrac{2a^2}{\sqrt{a}}+b\sqrt{b}}{a\sqrt{a}+b\sqrt{b}}+\dfrac{3\sqrt{ab}-3b}{a-b}$\\ | \\ $3$ | |
| | | \\ 10. | \\ $\left(\dfrac{1}{\sqrt{y}}-\dfrac{2}{\sqrt{x}+\sqrt{y}}\right):\left(\sqrt{x}-\dfrac{x+y}{\sqrt{x}+\sqrt{y}}\right)\cdot\sqrt{y}$\\ | \\ $\dfrac{1}{\sqrt{y}}$ | |
| | | \\ 11. | \\ $\left(\sqrt{c}+\sqrt{d}-\dfrac{2\sqrt{cd}}{\sqrt{c}+\sqrt{d}}\right):\left(\dfrac{\sqrt{c}-\sqrt{d}}{\sqrt{c}+\sqrt{d}}+\dfrac{\sqrt{d}}{\sqrt{c}}\right)\cdot\sqrt{c}$\\ | \\ $c$ | |
| | | \\ 12. | \\ $\left(\left(\sqrt{\dfrac{a}{b}}-\sqrt{\dfrac{b}{a}}\right):\left(\sqrt{\dfrac{a}{b}}+\sqrt{\dfrac{b}{a}}-2\right)\right):\left(1+\sqrt{\dfrac{b}{a}}\right)$ , при $a>0, b>0$\\ | \\ $\dfrac{\sqrt{a}}{\sqrt{a}-\sqrt{b}}$ | |
| | | \\ 13. | \\ $\left(\dfrac{1+\sqrt{1-x}}{1-x+\sqrt{1-x}}+\dfrac{1-\sqrt{1+x}}{1+x-\sqrt{1+x}}\right)^2\cdot\dfrac{x^2-1}{2}+1$ , при $0<x<1$\\ | \\ $\sqrt{1-x^2}$ | |
| | | \\ 14. | \\ $\left(\dfrac{\sqrt{a}+1}{\sqrt{ab}+1}+\dfrac{\sqrt{ab}+\sqrt{a}}{\sqrt{ab}-1}-1\right):\left(\dfrac{\sqrt{a}+1}{\sqrt{ab}+1}-\dfrac{\sqrt{ab}+\sqrt{a}}{\sqrt{ab}-1}+1\right)$\\ | \\ $-\sqrt{ab}$ | |
| | | \\ 15. | \\ $\left(\dfrac{2+\sqrt{a}}{a+2\sqrt{a}+1}-\dfrac{\sqrt{a}-2}{a-1}\right)\cdot\dfrac{a\sqrt{a}+a-\sqrt{a}-1}{\sqrt{a}}$\\ | \\ $2$ | |
| | | \\ 16. | \\ $\left(\dfrac{\sqrt{a}-\sqrt{b}}{a\sqrt{b}+b\sqrt{a}}+\dfrac{\sqrt{a}+\sqrt{b}}{a\sqrt{b}-b\sqrt{a}}\right)\cdot\dfrac{\left(\sqrt{a}\right)^3\cdot\sqrt{b}}{a+b}-\dfrac{2b}{a-b}$\\ | \\ $2$ | |
| | | \\ 17. | \\ $\sqrt{\dfrac{x}{x-a^2}}:\left(\dfrac{\sqrt{x}-\sqrt{x-a^2}}{\sqrt{x}+\sqrt{x-a^2}}-\dfrac{\sqrt{x}+\sqrt{x-a^2}}{\sqrt{x}-\sqrt{x-a^2}}\right)$\\ | \\ $\dfrac{a^2}{4(a^2-x)}$ | |
| | | \\ 18. | \\ $\left(\dfrac{1}{\sqrt{a}+\sqrt{a+b}}+\dfrac{1}{\sqrt{a}-\sqrt{a-b}}\right):\left(1+\sqrt{\dfrac{a+b}{a-b}}\right)$ | \\ $\dfrac{\sqrt{a-b}}{b}$ | |
| | | \\ 19. | \\ $\dfrac{a\left(\dfrac{\sqrt{a}+\sqrt{b}}{2b\sqrt{a}}\right)^{-1}+b\left(\dfrac{\sqrt{a}+\sqrt{b}}{2a\sqrt{b}}\right)^{-1}}{\left(\dfrac{a+\sqrt{ab}}{2ab}\right)^{-1}+\left(\dfrac{b+\sqrt{ab}}{2ab}\right)^{-1}}$\\ | \\ $\sqrt{ab}$ | |
| | | \\ 20. | \\ $\left(\dfrac{\sqrt{a}+\sqrt{x}}{\sqrt{a+x}}-\dfrac{\sqrt{a+x}}{\sqrt{a}+\sqrt{x}}\right)^{-2}-\left(\dfrac{\sqrt{a}-\sqrt{x}}{\sqrt{a+x}}-\dfrac{\sqrt{a+x}}{\sqrt{a}-\sqrt{x}}\right)^{-2}$\\ | \\ $\dfrac{a+x}{\sqrt{ax}}$ | |
| | | \\ 21. | \\ $\dfrac{x-y}{\sqrt{x}+\sqrt{y}}:\left(\left(\sqrt{\sqrt{x}}-\sqrt{\sqrt{y}}\right)^{-1}+\left(\sqrt{\sqrt{x}}+\sqrt{\sqrt{y}}\right)^{-1}\right)^{-2}$\\ | \\ $\dfrac{4\sqrt{x}}{\sqrt{x}-\sqrt{y}}$ | |
| | | \\ 22. | \\ $\left(\dfrac{1}{(\sqrt{a}+\sqrt{b})^{-2}}-\left(\dfrac{\sqrt{a}-\sqrt{b}}{a\sqrt{a}-b\sqrt{b}}\right)^{-1}\right):\sqrt{ab}$\\ | \\ $1$ | |
| | | \\ 23. | \\ $\sqrt{\left(\dfrac{\sqrt{a}+b}{\sqrt{a^3}-b^3}\right)^{-1}\cdot\dfrac{a-b^2}{a+b\sqrt{a}+b^2}}-\sqrt{a}$ , при $\sqrt{a}>b$\\ | \\ $-b$ | |
| | | \\ 24. | \\ $\left(\dfrac{\sqrt{a}-2}{a+2\sqrt{a}}+\dfrac{\sqrt{a}+2}{a-2\sqrt{a}}\right)\cdot\dfrac{(\sqrt{a})^3}{a+4}-\dfrac{8}{a-4}$\\ | \\ $2$ | |
| | | \\ 25. | \\ $\left(\dfrac{\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x-1}}+\dfrac{x-1}{\sqrt{x^2-1}-x+1}\right)\dfrac{1}{\sqrt{x^2-1}}$\\ | \\ $1$ | |
| | | \\ 26. | \\ $ $\\ | \\ $ $ | |
| | | \\ 27. (2.001 Сканави) | \\ $ \dfrac{\sqrt{x}+1}{x\sqrt{x}+x+\sqrt{x}}:\dfrac{1}{x^2-\sqrt{x}} $\\ | \\ $x-1 $ | |
| | | \\ 28. (2.002 Сканави) | \\ $ \left((\sqrt{\sqrt{p}}-\sqrt{\sqrt{q}})^{-2}+(\sqrt{\sqrt{p}}+\sqrt{\sqrt{q}})^{-2}\right):\dfrac{\sqrt{p}+\sqrt{q}}{p-q}$\\ | \\ $\dfrac{2(\sqrt{p}+\sqrt{q})^2}{p-q} $ | |
| | | \\ 29. (2.003 Сканави) | \\ $\dfrac{(\sqrt{a^2+a\sqrt{a^2-b^2}}-\sqrt{a^2-a\sqrt{a^2-b^2}})^2}{2a\sqrt{ab}}:\left(\sqrt{\dfrac{a}{b}}+\sqrt{\dfrac{b}{a}}-2\right);a>b>0 $\\ | \\ $\dfrac{(\sqrt{a}+\sqrt{b}^2)}{a-b} $ | |
| | | \\ 30. (2.006 Сканави) | \\ $\dfrac{(\sqrt{a}+\sqrt{b})^2-4b}{(a-b):\left(\sqrt{\dfrac{1}{b}}+3\sqrt{\dfrac{1}{a}}\right)}:\dfrac{a+9b+6\sqrt{ab}}{\sqrt{\dfrac{1}{b}}+\sqrt{\dfrac{1}{a}}}$\\ | \\ $\dfrac{1}{ab} $ | |
| | | \\ 31. (2.007 Сканави) | \\ $\dfrac{(\sqrt{\sqrt{m}}+\sqrt{\sqrt{n}})^2+(\sqrt{\sqrt{m}}-\sqrt{\sqrt{n}})^2}{2(m-n)}:\dfrac{1}{m\sqrt{m}-n\sqrt{n}}-3\sqrt{mn} $\\ | \\ $(\sqrt{m}-\sqrt{n})^2 $ | |
| | | \\ 32. (2.009 Сканави) | \\ $\dfrac{2\sqrt{1+\dfrac{1}{4}\left(\sqrt{\dfrac{1}{t}}-\sqrt{t}\right)^2}}{\sqrt{1+\dfrac{1}{4}\left(\sqrt{\dfrac{1}{t}}-\sqrt{t}\right)^2}-\dfrac{1}{2}\left(\sqrt{\dfrac{1}{t}}-\sqrt{t}\right)} $\\ | \\ $\dfrac{t+1}{t} $ | |
| | | \\ 33. (2.010 Сканави) | \\ $t\cdot\dfrac{1+\dfrac{2}{\sqrt{t+4}}}{2-\sqrt{t+4}}+\sqrt{t+4}+\dfrac{4}{\sqrt{t+4}} $\\ | \\ $-4 $ | |
| | | \\ 34. (2.011 Сканави) | \\ $\left(\dfrac{1+\sqrt{x}}{\sqrt{1+x}}-\dfrac{\sqrt{1+x}}{1+\sqrt{x}}\right)^2-\left(\dfrac{1-\sqrt{x}}{\sqrt{1+x}}-\dfrac{\sqrt{1+x}}{1-\sqrt{x}}\right)^2 $\\ | \\ $\dfrac{16x\sqrt{x}}{(1-x^2)(x-1)} $ | |
| | | \\ 35. (2.012 Сканави) | \\ $\dfrac{x-1}{x+\sqrt{x}+1}:\dfrac{\sqrt{x}+1}{x\sqrt{x}-1}+\dfrac{2}{\dfrac{1}{\sqrt{x}}} $\\ | \\ $x+1 $ | |
| | | \\ 36. (2.013 Сканави) | \\ $\left(\dfrac{1}{\sqrt{a}+\sqrt{a+1}}+\dfrac{1}{\sqrt{a}-\sqrt{a-1}}\right):\left(1+\dfrac{\sqrt{a+1}}{\sqrt{a-1}}\right) $\\ | \\ $\sqrt{a-1} $ | |
| | | \\ 37. (2.021 Сканави) | \\ $ \dfrac{4x(x+\sqrt{x^2-1})^2}{(x+\sqrt{x^2-1})^4-1} $\\ | \\ $\dfrac{1}{\sqrt{x^2-1}} $ | |
| | | \\ 38. (2.022 Сканави) | \\ $\dfrac{\sqrt{(x+2)^2-8x}}{\sqrt{x}-2:\sqrt{x}} $\\ | \\ $-\sqrt{x}$, при $x \in (0,2) $;\\ \\ $\sqrt{x}$, при $ x \in (2,+\infty) $ | |
| | | \\ 39. (2.028 Сканави) | \\ $\dfrac{x\cdot\dfrac{1}{\sqrt{x^2-a^2}}+1}{a\cdot\dfrac{1}{\sqrt{x-a}}+\sqrt{x-a}}:\dfrac{a^2\sqrt{x+a}}{x-\sqrt{x^2-a^2}}+\dfrac{1}{x^2-ax} $\\ | \\ $\dfrac{2}{x^2-a^2} $ | |
| | | \\ 40. (2.032 Сканави) | \\ $\dfrac{\sqrt{\dfrac{abc+4}{a}+4\sqrt{\dfrac{bc}{a}}}}{\sqrt{abc}+2}$ , при $a=0,04 $\\ | \\ $5 $ | |
| | | \\ 41. (2.033 Сканави) | \\ $\dfrac{(2p+1)\sqrt{2p+1}+(2p-1)\sqrt{2p-1}}{\sqrt{4p+2\sqrt{4p^2-1}}} $\\ | \\ $ 4p-\sqrt{4p^2-1}$ | |
| | | \\ 42. (2.034 Сканави) | \\ $1-\dfrac{\dfrac{1}{\sqrt{a-1}}-\sqrt{a+1}}{\dfrac{1}{\sqrt{a+1}}-\dfrac{1}{\sqrt{a-1}}}:\dfrac{\sqrt{a+1}\cdot\sqrt{a^2-1}}{(a-1)\sqrt{a+1}-(a+1)\sqrt{a-1}} $\\ | \\ $\sqrt{a^2-1} $ | |
| | | \\ 43. (2.037 Сканави) | \\ $\dfrac{1-\dfrac{1}{x^2}}{\sqrt{x}-\dfrac{1}{\sqrt{x}}}-\dfrac{2}{x\sqrt{x}}+\dfrac{\dfrac{1}{x^2}-x}{\sqrt{x}-\dfrac{1}{\sqrt{x}}} $\\ | \\ $-\sqrt{x}\left(1+\dfrac{2}{x^2}\right) $ | |
| | | \\ 44. (2.038 Сканави) | \\ $\left(\dfrac{\sqrt{a}}{2}-\dfrac{1}{2\sqrt{a}}\right)^2\left(\dfrac{\sqrt{a}-1}{\sqrt{a}+1}-\dfrac{\sqrt{a}+1}{\sqrt{a}-1}\right) $\\ | \\ $\dfrac{1-a}{\sqrt{a}} $ | |
| | | \\ 45. (2.041 Сканави) | \\ $\dfrac{1}{2(1+\sqrt{a})}+\dfrac{1}{2(1-\sqrt{a})}-\dfrac{a^2+2}{1-a^3} $\\ | \\ $-\dfrac{1}{a^2+a+1} $ | |
| | | \\ 46. (2.044 Сканави) | \\ $\left(\dfrac{\sqrt{x-a}}{\sqrt{x+a}+\sqrt{x-a}}+\dfrac{x-a}{\sqrt{x^2-a^2}-x+a}\right):\sqrt{\dfrac{x^2}{a^2}-1};x>a>0 $\\ | \\ $1 $ | |
| | | \\ 47. (2.046 Сканави) | \\ $\dfrac{\sqrt{1-x^2}-1}{x}\left(\dfrac{1-x}{\sqrt{1-x^2}+x-1}+\dfrac{\sqrt{1+x}}{\sqrt{1+x}-\sqrt{1-x}}\right) $\\ | \\ $-1 $ | |
| | | \\ 48. (2.052 Сканави) | \\ $\left(\dfrac{1}{\sqrt{1-x^2}}+1+\dfrac{1}{\dfrac{1}{\sqrt{1-x^2}}-1}\right)^{-2}:(2-x^2-2\sqrt{1-x^2}) $\\ | \\ $1-x^2 $ | |
| | | \\ 49. (2.053 Сканави) | \\ $\left(\dfrac{1}{\sqrt{1-p^2}}-\dfrac{1}{\sqrt{1+p^2}}\right)^2+\dfrac{2}{\sqrt{1-p^4}} $\\ | \\ $\dfrac{2}{1-p^4} $ | |
| | | \\ 50. (2.071 Сканави) | \\ $\dfrac{(m-1)\sqrt{m}-(n-1)\sqrt{n}}{\sqrt{m^3n}+mn+m^2-m} $\\ | \\ $\dfrac{\sqrt{m}-\sqrt{n}}{m} $ | |
| | | \\ 51. (2.079 Сканави) | \\ $\left(\sqrt{\sqrt{m}-\sqrt{\dfrac{m^2-9}{m}}}+\sqrt{\sqrt{m}+\sqrt{\dfrac{m^2-9}{m}}}\right)^2\sqrt{\sqrt{\dfrac{m^2}{4}}} $\\ | \\ $\sqrt{2}\cdot(m+3) $ | |
| | | \\ 52. (2.081 Сканави) | \\ $\sqrt{\dfrac{t\sqrt{t+2}}{\sqrt{t-2}}-\dfrac{2\sqrt{t-2}}{\sqrt{t+2}}-\dfrac{4t}{\sqrt{t^2-4}}}:\sqrt{\sqrt{t^2-4}} $\\ | \\ $\dfrac{\sqrt{t^2-4}}{t+2} $ | |
| | | \\ 53. (2.085 Сканави) | \\ $\left(\dfrac{a\sqrt{a}+b\sqrt{b}}{\sqrt{a}+\sqrt{b}}-\sqrt{ab}\right)\left(\dfrac{\sqrt{a}+\sqrt{b}}{a-b}\right)^2 $\\ | \\ $1 $ | |
| | | \\ 54. (2.086 Сканави) | \\ $\left(\dfrac{a-\sqrt{a^2-b^2}}{a+\sqrt{a^2-b^2}}-\dfrac{a+\sqrt{a^2-b^2}}{a-\sqrt{a^2-b^2}}\right):\dfrac{4\sqrt{a^4-a^2b^2}}{(5b)^2} $\\ | \\ $25$, при $a<0$\\ \\ $-25$, при $a>0$ | |
| | | \\ 55. (2.088 Сканави) | \\ $\left(\sqrt{1-x^2}+1\right):\left(\dfrac{1}{\sqrt{x+1}}+\sqrt{1-x}\right) $\\ | \\ $\sqrt{1+x} $ | |
| | | \\ 56. (2.090 Сканави) | \\ $\dfrac{\dfrac{(a-b)^3}{(\sqrt{a}+\sqrt{b})^3}+2a\sqrt{a}+b\sqrt{b}}{a\sqrt{a}+b\sqrt{b}}+\dfrac{3(\sqrt{ab}-b)}{a-b} $\\ | \\ $3 $ | |
| | | \\ 57. (2.093 Сканави) | \\ $\left(\dfrac{\sqrt{3}+1}{1+\sqrt{3}+\sqrt{t}}+\dfrac{\sqrt{3}-1}{1-\sqrt{3}+\sqrt{t}}\right)\left(\sqrt{t}-\dfrac{2}{\sqrt{t}}+2\right) $\\ | \\ $2\sqrt{3} $ | |
| | | \\ 58. (2.096 Сканави) | \\ $\sqrt{\dfrac{x}{x-a^2}}:\left(\dfrac{\sqrt{x}-\sqrt{x-a^2}}{\sqrt{x}+\sqrt{x-a^2}}-\dfrac{\sqrt{x}+\sqrt{x-a^2}}{\sqrt{x}-\sqrt{x-a^2}}\right) $\\ | \\ $\dfrac{a^2}{4(a^2-x)} $ | |
| | | \\ 59. (2.097 Сканави) | \\ $\dfrac{\left(\sqrt{x}+2\right)\left(\dfrac{2}{\sqrt{x}}-1\right)-\left(\sqrt{x}-2\right)\left(\dfrac{2}{\sqrt{x}}+1\right)-\dfrac{8}{\sqrt{x}}}{\left(2-\sqrt{x+2}\right):\left(\sqrt{\dfrac{2}{x}+1}-\dfrac{2}{\sqrt{x}}\right)} $\\ | \\ $2 $ | |
| | | \\ 60. (2.100 Сканави) | \\ $\dfrac{(z-z\sqrt{z}+2-2\sqrt{z})^2(1+\sqrt{z})^2}{z-2+\dfrac{1}{z}}-z\sqrt{z}\cdot\sqrt{\dfrac{4}{z}+4+z} $\\ | \\ $z(z+1)(z+2) $ | |
| | | \\ 61. (2.101 Сканави) | \\ $\left(\dfrac{1}{a+\sqrt{2}}-\dfrac{a^2+4}{a^3+2\sqrt{2}}\right):\left(\dfrac{a}{2}-\dfrac{1}{\sqrt{2}}+\dfrac{1}{a}\right)^{-1} $\\ | \\ $-\dfrac{\sqrt{2}}{2a} $ | |
| | | \\ 62. (2.103 Сканави) | \\ $(\sqrt{ab}-ab(a+\sqrt{ab})^{-1}):(2(\sqrt{ab}-b)(a-b)^{-1}) $\\ | \\ $0,5a $ | |
| | | \\ 63. (2.105 Сканави) | \\ $\left(\dfrac{1+\sqrt{1-x}}{1-x+\sqrt{1-x}}+\dfrac{1-\sqrt{1+x}}{1+x-\sqrt{1+x}}\right)^2\cdot\dfrac{x^2-1}{2}-\sqrt{1-x^2} $\\ | \\ $-1 $ | |
| | | \\ 64. (2.110 Сканави) | \\ $\dfrac{\sqrt{c-d}}{c^2\sqrt{2c}}\cdot\left(\sqrt{\dfrac{c-d}{c+d}}+\sqrt{\dfrac{c^2+cd}{c^2-cd}}\right)$ , при $c=2, d=\dfrac{1}{4} $\\ | \\ $\dfrac{1}{3} $ | |
| | | \\ 65. (2.115 Сканави) | \\ $4ab+\dfrac{\left(1+\left(\dfrac{a}{b}\right)^{-3}\right)a^3}{(\sqrt{a}+\sqrt{b})^2-2\sqrt{ab}}-\dfrac{\left(\dfrac{\sqrt{a}+\sqrt{b}}{2b\sqrt{a}}\right)^{-1}+\left(\dfrac{\sqrt{a}+\sqrt{b}}{2a\sqrt{b}}\right)^{-1}}{\left(\dfrac{a+\sqrt{ab}}{2}\right)^{-1}+\left(\dfrac{b+\sqrt{ab}}{2}\right)^{-1}} $\\ | \\ $(a+b)^2 $ | |
| | | \\ 66. (2.136 Сканави) | \\ $\dfrac{1-b}{\sqrt{b}}\cdot{x^2}-2x+\sqrt{b}$ , при $x=\dfrac{\sqrt{b}}{1-\sqrt{b}} $\\ | \\ $ 0$ | |
| | | \\ 67. (2.143 Сканави) | \\ $\dfrac{2b\sqrt{x^2-1}}{x-\sqrt{x^2-1}}$ , при $ x=\dfrac{1}{2}\left(\sqrt{\dfrac{a}{b}}+\sqrt{\dfrac{b}{a}}\right);a>b>0 $\\ | \\ $a-b $ | |
| | | \\ 68. (2.144 Сканави) | \\ $\dfrac{2a\sqrt{1+x^2}}{x+\sqrt{1+x^2}}$ , при $ x=\dfrac{1}{2}\left(\sqrt{\dfrac{a}{b}}-\sqrt{\dfrac{b}{a}}\right);a>0,b>0 $\\ | \\ $a+b $ | |
| | | \\ 69. (2.145 Сканави) | \\ $\dfrac{1-ax}{1+ax}\sqrt{\dfrac{1+bx}{1-bx}}$ , при $x=1\dfrac{1}{a}\sqrt{\dfrac{2a-b}{b}}; 0<\dfrac{b}{2}<a<b $\\ | \\ $1 $ | |
| | | \\ 70. (2.203 Сканави) | \\ $\dfrac{(2x+\sqrt{x^2-1})\cdot\sqrt{\sqrt{\dfrac{x-1}{x+1}}+\sqrt{\dfrac{x+1}{x-1}}-2}}{(x+1)\sqrt{x+1}-(x-1)\sqrt{x-1}} $\\ | \\ $\dfrac{1}{\sqrt{\sqrt{x^2-1}}} $ | |
| | | \\ 71. (2.210 Сканави) | \\ $\dfrac{2\sqrt{\dfrac{1}{4}\left(\dfrac{1}{\sqrt{a}}+\sqrt{a}\right)^2-1}}{2\sqrt{\dfrac{1}{4}\left(\dfrac{1}{\sqrt{a}}+\sqrt{a}\right)^2-1}-\dfrac{1}{2}\left(\sqrt{\dfrac{1}{a}-\sqrt{a}}\right)} $\\ | \\ $2$, при $a\in(0;1);$\\ \\ $\dfrac{2}{3}$, при $a\in(1;+\infty)$ | |
| | | \\ 72. (2.216 Сканави) | \\ $\left(\sqrt{\dfrac{m+2}{m-2}}+\sqrt{\dfrac{m-2}{m+2}}\right):\left(\sqrt{\dfrac{m+2}{m-2}}-\sqrt{\dfrac{m-2}{m+2}}\right) $\\ | \\ $0,5m $ | |
| | | \\ 73. (2.218 Сканави) | \\ $\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}} $\\ | \\ $2\sqrt{2} $ | |
| | | \\ 74. (2.233 Сканави) | \\ $\dfrac{\dfrac{1}{\sqrt{a-1}}-\sqrt{a+1}}{\dfrac{1}{\sqrt{a+1}}-\dfrac{1}{\sqrt{a-1}}}:\dfrac{\sqrt{a+1}}{(a-1)\sqrt{a+1}-(a+1)\sqrt{a-1}}-(1-a^2) $\\ | \\ $\sqrt{a^2-1} $ | |
| | | \\ 75. (2.236 Сканави) | \\ $\dfrac{\sqrt{z^2-1}}{\sqrt{z^2-1}-z}$ , при $z=\dfrac{1}{2}\left(\sqrt{m}+\dfrac{1}{\sqrt{m}}\right) $\\ | \\ $\dfrac{m-1}{2m}$, при $m \in(0;1)$\\ \\ $\dfrac{1-m}{m}$, при $m \in [1;+\infty)$ | |
| | | \\ 76. (2.279 Сканави) | \\ $\dfrac{a+b}{(\sqrt{a}-\sqrt{b})^2}\left(\dfrac{3ab-b\sqrt{ab}+a\sqrt{ab}-3b^3}{\dfrac{1}{2}\sqrt{\dfrac{1}{4}\left(\dfrac{a}{b}+\dfrac{b}{a}\right)^2-1}}+\dfrac{4ab\sqrt{a}+9ab\sqrt{b}-9b^2\sqrt{a}}{\dfrac{3}{2}\sqrt{b}-2\sqrt{a}}\right)$ , при $a>b>0 $\\ | \\ $-2b(a+3\sqrt{ab}) $ | |
| | | \\ 77. (2.280 Сканави) | \\ $\dfrac{2a(a+2b+\sqrt{a^2+4ab})}{(a+\sqrt{a^2+4ab})(a+4b+\sqrt{a^2+4ab})} $\\ | \\ $\sqrt{\dfrac{a}{a+4b}} $ | |
| | | \\ 78. (2.282 Сканави) | \\ $\left(\dfrac{\sqrt{1+x}}{\sqrt{1+x}-\sqrt{1-x}}+\dfrac{1-x}{\sqrt{1-x^2}-1+x}\right)\left(\sqrt{\dfrac{1}{x^2}-1}-\dfrac{1}{x}\right)$ , при $0<x<1 $\\ | \\ $-1 $ | |
| |
