| Предыдущая версия справа и слеваПредыдущая версияСледующая версия | Предыдущая версия |
| math-public:uproscheniya_s_bukvami [2020/01/17 11:34] – labreslav | math-public:uproscheniya_s_bukvami [2020/01/17 23:30] (текущий) – labreslav |
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| | \\ 26. | \\ $ $\\ | \\ $ $ | | | \\ 26. | \\ $ $\\ | \\ $ $ | |
| | \\ 27. (2.001 Сканави) | \\ $ \dfrac{\sqrt{x}+1}{x\sqrt{x}+x+\sqrt{x}}:\dfrac{1}{x^2-\sqrt{x}} $\\ | \\ $x-1 $ | | | \\ 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} $ | | | \\ 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} $ | | | \\ 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} $ | | | \\ 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)} $ | |
| | \\ 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 $ | | | \\ 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 $ | | | \\ 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}}-\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} $ | | | \\ 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) $ | |
| | \\ 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} $ | |
| | \\ 62. (2.103 Сканави) | \\ $(\sqrt{ab}-ab(a+\sqrt{ab})^{-1}):(2(\sqrt{ab}-b)(a-b)^{-1}) $\\ | \\ $0,5a $ | | | \\ 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 $ | | | \\ 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} $ | | | \\ 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 $ | | | \\ 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 $ | |
| | \\ 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}}} $ | | | \\ 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 $a\in(1;+\infty)$ | | | \\ 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 $ | | | \\ 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} $ | | | \\ 73. (2.218 Сканави) | \\ $\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}} $\\ | \\ $2\sqrt{2} $ | |
