| Предыдущая версия справа и слеваПредыдущая версияСледующая версия | Предыдущая версияСледующая версияСледующая версия справа и слева |
| math-public:uproscheniya_s_bukvami [2020/01/16 18:15] – labreslav | math-public:uproscheniya_s_bukvami [2020/01/17 11:51] – labreslav |
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| | \\ 16. | \\ $\left(\dfrac{\sqrt{a}-\sqrt{b}}{a\sqrt{b}+b\sqrt{a}}+\dfrac{\sqrt{a}+\sqrt{b}}{a\sqrt{b}-b\sqrt{a}}\right)\cdot\dfrac{\left(\sqrt{a}\right)^3\cdot\sqrt{b}}{a+b}-\dfrac{2b}{a-b}$\\ | \\ $2$ | | | \\ 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. | \\ $\sqrt{\dfrac{x}{x-a^2}}:\left(\dfrac{\sqrt{x}-\sqrt{x-a^2}}{\sqrt{x}+\sqrt{x-a^2}}-\dfrac{\sqrt{x}+\sqrt{x-a^2}}{\sqrt{x}-\sqrt{x-a^2}}\right)$\\ | \\ $\dfrac{a^2}{4(a^2-x)}$ | | | \\ 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. | \\ $\left(\dfrac{1}{\sqrt{a}-\sqrt{a+b}}-\dfrac{1}{\sqrt{a}+\sqrt{a-b}}\right):\left(1+\sqrt{\dfrac{a+b}{a-b}}\right)$\\ | \\ $\dfrac{\sqrt{a-b}}{b}$ | | | \\ 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. | \\ $\dfrac{a\left(\dfrac{\sqrt{a}+\sqrt{b}}{2b\sqrt{a}}\right)^{-1}+b\left(\dfrac{\sqrt{a}+\sqrt{b}}{2a\sqrt{b}}\right)^{-1}}{\left(\dfrac{a+\sqrt{ab}}{2ab}\right)^{-1}+\left(\dfrac{b+\sqrt{ab}}{2ab}\right)^{-1}}$\\ | \\ $\sqrt{ab}$ | | | \\ 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. | \\ $\left(\dfrac{\sqrt{a}+\sqrt{x}}{\sqrt{a+x}}-\dfrac{\sqrt{a+x}}{\sqrt{a}+\sqrt{x}}\right)^{-2}-\left(\dfrac{\sqrt{a}-\sqrt{x}}{\sqrt{a+x}}-\dfrac{\sqrt{a+x}}{\sqrt{a}-\sqrt{x}}\right)^{-2}$\\ | \\ $\dfrac{a+x}{\sqrt{ax}}$ | | | \\ 20. | \\ $\left(\dfrac{\sqrt{a}+\sqrt{x}}{\sqrt{a+x}}-\dfrac{\sqrt{a+x}}{\sqrt{a}+\sqrt{x}}\right)^{-2}-\left(\dfrac{\sqrt{a}-\sqrt{x}}{\sqrt{a+x}}-\dfrac{\sqrt{a+x}}{\sqrt{a}-\sqrt{x}}\right)^{-2}$\\ | \\ $\dfrac{a+x}{\sqrt{ax}}$ | |
| | \\ 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} $ | | | \\ 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 $ | | | \\ 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} $ | | | \\ 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 $ | | | \\ 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)} $ | | | \\ 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)} $ | |
| | \\ 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 $ | | | \\ 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}}-\sqrt{1+p^2}\right)^2+\dfrac{2}{\sqrt{1-p^4}} $\\ | \\ $\dfrac{2}{1-p^4} $ | | | \\ 49. (2.053 Сканави) | \\ $\left(\dfrac{1}{\sqrt{1-p^2}}-\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}}{m^2\sqrt{mn}+mn+m^2-m} $\\ | \\ $\dfrac{\sqrt{m}-\sqrt{n}}{m} $ | | | \\ 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) $ | | | \\ 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} $ | | | \\ 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 $ | | | \\ 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 $a>0$ | | | \\ 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{x-1}\right) $\\ | \\ $\sqrt{1+x} $ | | | \\ 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 $ | | | \\ 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} $ | | | \\ 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)} $ | | | \\ 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}-\sqrt{x}-2\right)\left(\dfrac{2}{\sqrt{x}+1}-\dfrac{8}{\sqrt{x}}\right)}{\left(2-\sqrt{x+2}\right):\left(\sqrt{\dfrac{2}{x}+1}-\dfrac{2}{\sqrt{x}}\right)} $\\ | \\ $2 $ | | | \\ 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) $ | | | \\ 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} $ | | | \\ 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} $ | |
| | \\ 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 $ | | | \\ 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$ | | | \\ 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>0,b>0 $\\ | \\ $a-b $ | | | \\ 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 $ | | | \\ 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 $ | | | \\ 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 $ | |
