#1
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àŢ¡¡ÓÅѧ
ªèÇÂ˹è͹ФÃѺ
1. $\frac{4+2\sqrt{3}}{\sqrt[3]{10+6\sqrt{3}}}+\sqrt{3}$ à·èҡѺà·èÒã´ \sqrt[n]{x} 2. $\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{4}}+...+\frac{1}{\sqrt{1023}+\sqrt{1024}}$ ÁÕ¤èÒà·èÒã´ 3. $2^x = 3^y = 4^z = 24^{10}$ áÅéÇ $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ ÁÕ¤èÒà·èÒã´ 4.$4^{2-x}+2^{3-2x}+2^{2-2x}=14$ ¨§ËÒ¤èÒ x 5.¨§àÃÕ§ÅӴѺ¨Ò¡¨Ó¹Ç¹¹éÍÂä»ÁÒ¡ A=$2^{2^{2^{2^2}}}$ $B=2^{5^{2^{1^{9^7}}}}$ $C=4^{{10}^5}-4^{\sqrt[3]{1331}}$ $D=4^{2552}$ 6. $\frac{x^{-1}}{x^{-1}+y^{-1}}$ ·ÓäÁ¨Ö§à·èҡѺ $\frac{y}{x+y}$ ÎÐ ¢ÍÇÔ¸Õ·Ó´éǹФÃѺ ¢Íº¤Ø³¤ÃѺ 05 ¡Ã¡®Ò¤Á 2011 20:31 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 7 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ [T]ime[Z]ero |
#2
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1. ⨷ÂìäÁè¼Ô´ãªèäËÁ
2. Conjugate 3. ¨Ñº¤ÙèÊÁ¡Òà 4. ÊѧࡵÇèÒÁÕÍÐäÃàËÁ×͹¡Ñ¹ 5. ... 6. àÈÉÊèǹ«é͹ |
#3
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ÍéÒ§ÍÔ§:
$ = \frac{1-\sqrt{2}}{1-2} + \frac{\sqrt{2}-\sqrt{3}}{2-3} + \frac{\sqrt{3}-\sqrt{4}}{3-4} + ... + \frac{\sqrt{1023} -\sqrt{1024} }{1023-1024}$ $ = \frac{1-\sqrt{2}}{-1} + \frac{\sqrt{2}-\sqrt{3}}{-1} + \frac{\sqrt{3}-\sqrt{4}}{-1} + ... + \frac{\sqrt{1023} -\sqrt{1024} }{-1}$ $ = -(1-\sqrt{2} ) - (\sqrt{2} -\sqrt{3} ) - (\sqrt{3} -\sqrt{4}) - ... - (\sqrt{1023} -\sqrt{1024} ) $ $ = \sqrt{2} -1 +\sqrt{3} -\sqrt{2} +\sqrt{4} -\sqrt{3} + ... + \sqrt{1024}- \sqrt{1023} $ $ \sqrt{1024} -1 = 32 - 1 = 31$
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ÁÒËÒ¤ÇÒÁÃÙéäÇéµÔÇËÅÒ¹ áµèËÅÒ¹äÁèàÍÒàÅ¢áÅéÇ à¢éÒÁÒ·ÓàÅ¢àÍÒÁѹÍÂèÒ§à´ÕÂÇ ¤ÇÒÁÃÙéà»ç¹ÊÔè§à´ÕÂÇ·ÕèÂÔè§ãËé ÂÔè§ÁÕÁÒ¡ ÃÙéÍÐäÃäÁèÊÙé ÃÙé¨Ñ¡¾Í (¡àÇ鹤ÇÒÁÃÙé äÁèµéͧ¾Í¡çä´é ËÒäÇéÁÒ¡æáËÅдÕ) (áµè¡çÍÂèÒãËéÁÒ¡¨¹·èÇÁËÑÇ àÍÒµÑÇäÁèÃÍ´) 04 ¡Ã¡®Ò¤Á 2011 22:40 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ banker à˵ؼÅ: á¡é䢵ÒÁ¤Óá¹Ð¹Ó¢Í§¤Ø³Amankris |
#4
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ÍéÒ§ÍÔ§:
$ 2 = 24^\frac{10}{x}$ ...(1) $3^y = 24^{10}$ $ 3 = 24^\frac{10}{y}$ ...(2) $4^z = 24^{10}$ $ 4 = 24^\frac{10}{z}$ ...(3) (1)x(2)x(3) $ \ \ \ 2 \times3 \times4 = 24 ^{10(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})}$ $24^1 = 24 ^{10(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})}$ $\frac{1}{x}+\frac{1}{y}+\frac{1}{z} = \frac{1}{10}$
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ÁÒËÒ¤ÇÒÁÃÙéäÇéµÔÇËÅÒ¹ áµèËÅÒ¹äÁèàÍÒàÅ¢áÅéÇ à¢éÒÁÒ·ÓàÅ¢àÍÒÁѹÍÂèÒ§à´ÕÂÇ ¤ÇÒÁÃÙéà»ç¹ÊÔè§à´ÕÂÇ·ÕèÂÔè§ãËé ÂÔè§ÁÕÁÒ¡ ÃÙéÍÐäÃäÁèÊÙé ÃÙé¨Ñ¡¾Í (¡àÇ鹤ÇÒÁÃÙé äÁèµéͧ¾Í¡çä´é ËÒäÇéÁÒ¡æáËÅдÕ) (áµè¡çÍÂèÒãËéÁÒ¡¨¹·èÇÁËÑÇ àÍÒµÑÇäÁèÃÍ´) |
#5
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ÍéÒ§ÍÔ§:
$ = 2 ^{2(2-x)}+2^{3-2x}+2^{2-2x}=14$ $\frac{16}{2^{2x}} + \frac{8}{2^{2x}} + \frac{4}{2^{2x}} = 14 $ $\frac{2}{2^{2x}} = 1$ $2^1 = 2^{2x}$ $x= \frac{1}{2}$
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ÁÒËÒ¤ÇÒÁÃÙéäÇéµÔÇËÅÒ¹ áµèËÅÒ¹äÁèàÍÒàÅ¢áÅéÇ à¢éÒÁÒ·ÓàÅ¢àÍÒÁѹÍÂèÒ§à´ÕÂÇ ¤ÇÒÁÃÙéà»ç¹ÊÔè§à´ÕÂÇ·ÕèÂÔè§ãËé ÂÔè§ÁÕÁÒ¡ ÃÙéÍÐäÃäÁèÊÙé ÃÙé¨Ñ¡¾Í (¡àÇ鹤ÇÒÁÃÙé äÁèµéͧ¾Í¡çä´é ËÒäÇéÁÒ¡æáËÅдÕ) (áµè¡çÍÂèÒãËéÁÒ¡¨¹·èÇÁËÑÇ àÍÒµÑÇäÁèÃÍ´) |
#6
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ÍéÒ§ÍÔ§:
$\frac{x^{-1}}{x^{-1}+y^{-1}}$ $ = \dfrac{\frac{1}{x}}{\frac{1}{x}+\frac{1}{y}}$ $ = \dfrac{\frac{1}{x}} {\frac{x+y}{xy}}$ $ = \frac{1}{x} \times \frac{xy}{x+y}$ $ = \frac{y}{x+y}$
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ÁÒËÒ¤ÇÒÁÃÙéäÇéµÔÇËÅÒ¹ áµèËÅÒ¹äÁèàÍÒàÅ¢áÅéÇ à¢éÒÁÒ·ÓàÅ¢àÍÒÁѹÍÂèÒ§à´ÕÂÇ ¤ÇÒÁÃÙéà»ç¹ÊÔè§à´ÕÂÇ·ÕèÂÔè§ãËé ÂÔè§ÁÕÁÒ¡ ÃÙéÍÐäÃäÁèÊÙé ÃÙé¨Ñ¡¾Í (¡àÇ鹤ÇÒÁÃÙé äÁèµéͧ¾Í¡çä´é ËÒäÇéÁÒ¡æáËÅдÕ) (áµè¡çÍÂèÒãËéÁÒ¡¨¹·èÇÁËÑÇ àÍÒµÑÇäÁèÃÍ´) |
#7
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#3
àªç¤Ë¹èͤÃѺ |
#8
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¢Íº¤Ø³¤ÃѺ á¡éä¢áÅéǤÃѺ
(¢éͼԴ¾ÅÒ´ ... ¡Ò÷ӢéÒÁ¢Ñ鹵͹ ·ÓãËé¼Ô´¾ÅÒ´ä´é§èÒÂ)
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ÁÒËÒ¤ÇÒÁÃÙéäÇéµÔÇËÅÒ¹ áµèËÅÒ¹äÁèàÍÒàÅ¢áÅéÇ à¢éÒÁÒ·ÓàÅ¢àÍÒÁѹÍÂèÒ§à´ÕÂÇ ¤ÇÒÁÃÙéà»ç¹ÊÔè§à´ÕÂÇ·ÕèÂÔè§ãËé ÂÔè§ÁÕÁÒ¡ ÃÙéÍÐäÃäÁèÊÙé ÃÙé¨Ñ¡¾Í (¡àÇ鹤ÇÒÁÃÙé äÁèµéͧ¾Í¡çä´é ËÒäÇéÁÒ¡æáËÅдÕ) (áµè¡çÍÂèÒãËéÁÒ¡¨¹·èÇÁËÑÇ àÍÒµÑÇäÁèÃÍ´) |
#9
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ÍéÒ§ÍÔ§:
ÁÒ´ÙãËÁè ¨¢¡·. á¡éä¢â¨·ÂìãËÁèáÅéÇ $\frac{4+2\sqrt{3}}{\sqrt[3]{10+6\sqrt{3}}}+\sqrt{3}$ $= \frac{4+2\sqrt{3}}{\sqrt[3]{(1+\sqrt{3} )^3} } + \sqrt{3} $ $ = \frac{2(2+\sqrt{3} )}{1+\sqrt{3} } +\sqrt{3} $ $ = \frac{2(2+\sqrt{3} )(1-\sqrt{3} )}{1-3 } +\sqrt{3} $ $ = -(2-2\sqrt{3} +\sqrt{3}-3 ) +\sqrt{3} $ $ = -2+2\sqrt{3} -\sqrt{3}+3+\sqrt{3} $ $ = 1+2\sqrt{3} $
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ÁÒËÒ¤ÇÒÁÃÙéäÇéµÔÇËÅÒ¹ áµèËÅÒ¹äÁèàÍÒàÅ¢áÅéÇ à¢éÒÁÒ·ÓàÅ¢àÍÒÁѹÍÂèÒ§à´ÕÂÇ ¤ÇÒÁÃÙéà»ç¹ÊÔè§à´ÕÂÇ·ÕèÂÔè§ãËé ÂÔè§ÁÕÁÒ¡ ÃÙéÍÐäÃäÁèÊÙé ÃÙé¨Ñ¡¾Í (¡àÇ鹤ÇÒÁÃÙé äÁèµéͧ¾Í¡çä´é ËÒäÇéÁÒ¡æáËÅдÕ) (áµè¡çÍÂèÒãËéÁÒ¡¨¹·èÇÁËÑÇ àÍÒµÑÇäÁèÃÍ´) 04 ¡Ã¡®Ò¤Á 2011 23:07 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ banker |
#11
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ÍéÒ§ÍÔ§:
$B=2^{5^{2^{1^{9^7}}}} = 2^{5^2} = 2^{25}$ $C=4^{10^{5-4^{\sqrt[3]{1331} }}} = 4^{10^{(5-4^{11})}} = 4^{10^{µÔ´Åº}} = \frac{1}{4^{10^{à·èÒäÃäÁèÃÙé}}}$ $D = 4^{2552} = (2^2)^{2552} = 2^{5104}$ àÃÕ§¨Ò¡¹éÍÂä»ÁÒ¡ C, B, D, A
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ÁÒËÒ¤ÇÒÁÃÙéäÇéµÔÇËÅÒ¹ áµèËÅÒ¹äÁèàÍÒàÅ¢áÅéÇ à¢éÒÁÒ·ÓàÅ¢àÍÒÁѹÍÂèÒ§à´ÕÂÇ ¤ÇÒÁÃÙéà»ç¹ÊÔè§à´ÕÂÇ·ÕèÂÔè§ãËé ÂÔè§ÁÕÁÒ¡ ÃÙéÍÐäÃäÁèÊÙé ÃÙé¨Ñ¡¾Í (¡àÇ鹤ÇÒÁÃÙé äÁèµéͧ¾Í¡çä´é ËÒäÇéÁÒ¡æáËÅдÕ) (áµè¡çÍÂèÒãËéÁÒ¡¨¹·èÇÁËÑÇ àÍÒµÑÇäÁèÃÍ´) |
#12
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¢Íº¤Ø³ÁÒ¡¤ÃѺ
¢éÍ 5 $C=4^{{10}^5} - 4^{\sqrt[3]{1331}}$ ÎÐ ¢éÍ 1 ÂѧäÁè¤èÍÂà¢éÒ㨠·ÓäÁ $\sqrt[3]{10+6\sqrt{3}}=1+\sqrt{3}$ àËÃͤÃѺ ¶éÒ ËÃÁ. ¢Í§ $x^2-\frac{1}{x^2} , x^3-\frac{1}{x^3}$ áÅÐ $x^4-\frac{1}{x^4} = 3$ áÅéÇ $x^3-\frac{1}{x^3}$ ÁÕ¤èÒà·èÒã´ |
#13
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#12
ä»á¡é¢éÍ 5 ã¹ #1 ãËéÁѹÍèÒ¹ÃÙéàÃ×èͧä´éäËÁ ¢éÍ 1 Åͧ¡ÃШÒ´٠¤Ó¶ÒÁãËÁè ÅͧËÒ ËÃÁ. ÁÒ¡è͹ |
#14
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ËÒ ËÃÁ. Âѧä§àËÃͤÃѺ
---------------------------- $\frac{(\sqrt[3]{-8}+\sqrt{0.64})^3(\sqrt[3]{0.125})}{[(-2)^2x\sqrt[3]{216}]^2}$ ¢é͹Õé¼Á·Óä´é -0.0015 ¹èФÃѺ áµèäÁèÁÕ㹪éÍ ¶Ù¡ËÃ×Íà»ÅèÒÎÐ 1. 0.0012 2. -0.0012 3. 0.0015 4. -0.15 ----------------------------- $(1-a^{\frac{1}{32}})(1+a^{\frac{1}{32}})(1+a^{\frac{1}{16}})(1+a^{\frac{1}{8}})(1+a^{\frac{1}{4}})(1+a^{\frac{1}{2}})(1+a)$ ÁÕ¤èÒà·èÒã´ |
#15
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#14
ᡵÑÇ»ÃСͺ¤ÃѺ ÅͧáÊ´§ÇÔ¸ÕÁҹФÃѺ ¤Ù³µÃ§æàŤÃѺ |
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