第九章數的開方

(1)

注意：

允許學生個人、非營利性的圖書館或公立學校合理使用

本基金會網站所提供之各項試題及其解答。可直接下載

而不須申請。

重版、系統地複製或大量重製這些資料的任何部分，必

須獲得財團法人臺北市九章數學教育基金會的授權許

可。

申請此項授權請電郵

ccmp@seed.net.tw

Notice:

Individual students, nonprofit libraries, or schools are

permitted to make fair use of the papers and its

solutions. Republication, systematic copying, or

multiple reproduction of any part of this material is

permitted only under license from the Chiuchang

Mathematics Foundation.

Requests for such permission should be made by

(2)

ϫ!!ᐂ!

!

ಃΐക ኧޑ໒Б

……….……… 1 ߕᒵ ѳБਥޑ฽ᆉ؃ݤ………. 28

ಃΜക ΒԛਥԄ

………...……… 33

ಃΜ΋ക ΋ϡΒԛБำ

……...………...……… 67 ΋ ΋ϡΒԛБำ……… 67 Β ΋ϡΒԛБำਥᆶ߯ኧޑᜢ߯……… 87 Ο ёϯࣁ΋ϡΒԛБำޑБำ……… 95 Ѥ ᙁൂΒϡΒԛБำಔ……… 108

ಃΜΒക ࡰኧ

……...………...……… 127

ಃΜΟക தҔჹኧ

.………...……… 153

ಃΜѤക ڄኧϷځკ׎

……...………...……… 171 ΋ ޔف০኱س……… 171 Β ڄኧ……… 180 Ο ҅КٯڄኧᆶϸКٯڄኧ……… 188 Ѥ ΋ԛڄኧޑკ׎ᆶ܄፦……… 196 ϖ Βԛڄኧޑკ׎ᆶ܄፦……… 203 Ϥ ΋ϡ΋ԛό฻Ԅಔᆶ΋ϡΒԛό฻Ԅ……… 212

ಃΜϖക ှΟفڄኧ

……...…………...……… 229 ΋ Οفڄኧ……… 229 Β ှޔفΟف׎……… 241 Ο ှ௹Οف׎……… 252

(3)

ಃΜϤക ಍ी߃؁

.………...……… 275

ߕ ᒵ ߃ύжኧᕴፄಞୖԵᚒ

……… 306

!

(4)

ᄲ!!ځ!

ᶨˣġ 㛔㚠⊭㊔烉㔠䘬攳㕡ˣḴ㫉㟡⺷ˣᶨ⃫Ḵ㫉㕡䦳ˣ㊯㔠ˣ ⷠ䓐⮵㔠ˣ↥㔠⍲℞⚾⼊ˣ妋ᶱ奺↥㔠ˣ䴙妰⇅㬍炻ὃℓ ⸜䳂ᶲ⬠㛇冯ḅ⸜䳂ᶲ⬠㛇ἧ䓐ˤ Ḵˣġ 㛔㚠䘬佺柴ℙ↮ᶱ栆烉䶜佺ˣ佺柴ˣ墯佺⍫侫柴ˤ (1) ግಞ ὃ婚ℏ䶜佺ἧ䓐ˤ (2) ಞᚒ ὃ婚ℏ婚⢾ἄ㤕怠䓐ˤ (3) ፄಞୖԵᚒ ὃ㭷䪈墯佺怠䓐炻℞ᷕ⮹慷ⷞ㚱ˬ*˭嘇䘬柴⁭ὃ⬠㚱检≃䘬⬠䓇⍫侫ἧ䓐ˤ ᶱˣġ 㛔⣿㚠㒂⣏映⛘⋨ 1984 䘤Ựᷳ⇅䳂ᷕ⬠婚㛔㓡䶐ˤ

тңֹϡώ३ҋጯ!

1. ⃰徸⫿徸⎍教嬨⎬䪈䭨ᷳ妋婒ˤ 2. 忯㚱ἳ柴⛐嬨ㅪᷳ⼴炻ℵ⮯㚠㛔⎰ᶲ炻⎎冒埴⛐䫮姀㛔ᶲ㧉㒔妋㱽ˤ 3. ⛐䫮姀㛔ᶲ⮓⎬䪈䭨ᷳ䶜佺ˣ佺柴ˣ墯佺⍫侫柴ᷳ娛妋炻⎗ ᶵ㈬柴䚖炻Ữ天㧁㖶柴嘇冯枩䡤炻㭷ᶨ柴悥⽭枰娛䳘⮓↢㺼䬿忶䦳冯゛㱽ˤ 4. 㭷ᶨ柴 ⬴⼴⽭枰梲ㆸ䩳⌛槿䬿䘬佺ㄋ炻槿䬿䃉婌⼴⛐柴嘇 ἄˬΉ˭ᷳ姀嘇ˤ 5. 忯㚱ᶵ㚫ἄㆾᶵ㖶䡢柴シ䘬⓷柴炻婳⚆柕慵墯教嬨㬌䪈䭨ᷳ 妋婒炻䚜⇘⬴ℐ㖶䘥䁢㬊ˤ 6. ⎗䳬䷼ 4-6 Ṣᷳ嬨㚠㚫炻䶐⭂忚⹎炻⭂㛇倂㚫Ḻ䚠妶婾ˤ 7. 㛔⣿㚠⎗⛐ᶱᾳ㚰ℏ(ℙ䲬屣㗪 200 ⮷㗪)⬴ℐ⬠⤥⇅ᷕℐ悐㔠⬠婚䦳炻℞ᷕẋ㔠 1~4 Ⅎ冯⸦ỽ 1~2 Ⅎ⎗Ἕ埴䞼嬨ˤ

(5)

ຏǺ㛔㚠ᷕ㇨ἧ䓐ᷳ⹎慷堉╖ỵ㚱

ߏࡋൂՏ烉 km (kilometer炻℔慴)ˣm (meter炻℔⯢)ˣ cm (centimeter炻℔↮)ˣmm (millmeter炻℔慸)烊 1 km = 1000 m炻1 m = 100 cm = 1000 mm烊 ख़ໆൂՏ烉 T (Tonne炻℔☠)ˣkg (kilogram炻℔㕌)ˣ g (gram炻℔⃳)烊 1 T = 1000 kg炻1 kg = 1000 g烊 ৒ໆൂՏ烉 kL (kiloliter炻℔䥱)ˣL (liter炻℔⋯)ˣ mL (milliliter炻㮓⋯)烊 1 kL = 1000 L炻1 L = 1000 mLˤ

(6)

ௐ

˝

ౢ

!

ᇴ

۞

ฟ

͞!

͞

9.1 π͞ॲ

⤪㝄⶚䴻䞍忻ᶨᾳ㬋㕡⼊䘬怲攟炻ㆹᾹ⎗ẍ忂忶⸛㕡忳䬿㯪 ↢㬋㕡⼊䘬朊䧵ˤ⍵忶Ἦ炻⤪㝄⶚䞍㬋㕡⼊䘬朊䧵炻忁ᾳ㬋㕡⼊䘬怲攟ㅱ娚⾶㧋妰䬿烎 ἳ⤪炻ㆹᾹ天 ᶨᾳ朊䧵㗗 9 m2 䘬㕡㟴朊炻⯙天⃰㯪↢忁ᾳ 㕡㟴朊䘬怲攟㗗⣂⮹ m炻ḇ⯙㗗天㯪↢ᶨᾳ⸛㕡⼴䫱㕤 9 䘬㔠ˤ ⚈䁢 2 3 = 9炻_{( 3)}− 2 = ₉炻侴⮎晃ᶲ㟴朊䘬怲攟ᶵ傥㗗屈㔠炻㇨ẍ炻忁ᾳ㕡㟴朊䘬怲攟㗗 3 mˤ ᶨ凔⛘炻⤪㝄ᶨᾳ㔠䘬⸛㕡䫱㕤 a炻恋忁ᾳ㔠⯙⎓ a 䘬ѳ Бਥ(ḇ⎓ ΒԛБਥ)ˤ㎃⎍娙婒炻⤪㝄 2 x = a炻恋湤炻x ⯙⎓ a 䘬⸛㕡㟡ˤἳ⤪炻⚈䁢32 = 9炻_{( 3)}− 2 = ₉炻㇨ẍ 3 冯 3− 悥㗗 9 䘬⸛㕡㟡ˤ⍰⤪炻⚈䁢 2 7 = 49炻_{( 7)}− 2 = ₄₉炻㇨ẍ 7 冯 7− 悥㗗 49 䘬⸛㕡㟡ˤ栆Ụ⛘炻⚈䁢 2 4 2 25 5 = 炻 2 4 2 25 5 = − 炻㇨ẍ 2 5 冯 2 5 − 悥㗗 4 25䘬⸛㕡㟡ˤ 㟡㒂⸛㕡忳䬿⎗ẍ䞍忻炻ᶵ䁢暞䘬ℑᾳ㔠Ḻ䁢䚠⍵㔠㗪炻⬫ Ᾱ䘬⸛㕡⺷⎴ᶨᾳ㬋㔠ˤᶨ凔⛘炻΋ঁ҅ኧԖٿঁѳБਥǴ೭ٿ ঁѳБਥϕࣁ࣬ϸኧˤ ⚈䁢02 = 0炻㇨ẍ႟ޑѳБਥࢂ႟ˤ ảỽ㬋㔠ˣ屈㔠䘬⸛㕡悥㗗㬋㔠炻暞䘬⸛㕡㗗暞炻ḇ⯙㗗㬋㔠ˣ屈㔠冯暞䘬⸛㕡悥ᶵ㗗屈㔠炻⚈㬌ॄኧؒԖѳБਥ炻ἳ⤪炻 4 − 㰺㚱⸛㕡㟡ˤ 㯪ᶨᾳ㔠 a 䘬⸛㕡㟡ᷳ忳䬿炻⎓ ໒ѳБˤ 攳⸛㕡冯⸛㕡Ḻ䁢微忳䬿炻⚈㬌炻ㆹᾹ⎗ẍ忳䓐⸛㕡忳䬿㯪 ᶨᾳ㔠䘬⸛㕡㟡炻ḇ⎗ẍ䓐⸛㕡忳䬿㩊槿ᶨᾳ㔠㗗ᶵ㗗⎎ᶨᾳ㔠䘬⸛㕡㟡ˤἳ⤪炻

(7)

2 3 = 9 炻_{( 3)}− 2 = ₉ ∴ 9 䘬ℑᾳ⸛㕡㟡㗗 3 冯 3− ˤ ᶨᾳ㬋㔠 a 䘬㬋⸛㕡㟡炻䓐䫎嘇ˬ 2 a ˭堐䣢烊屈䘬⸛㕡㟡炻 䓐䫎嘇ˬ 2 a − ˭堐䣢ˤ忁ℑᾳ⸛㕡㟡⎰崟Ἦ⎗ẍ姀 ˬ±2 _a ˭ˤ 忁墉䫎嘇ˬ 2 _{˭嬨ἄˬḴ㫉㟡嘇˭炻a ⎓ ೏໒Бኧ炻2 ⎓ ਥ} ࡰኧˤ㟡㊯㔠㗗 2 㗪炻忂ⷠ䚩䔍ᶵ⮓炻⤪ 2 a ± ⮓ἄ± _a炻嬨ἄˬ㬋ˣ 屈㟡嘇 a˭ˤ± 0 ⯙㗗 0ˤ ݙཀǺ⚈䁢屈㔠㰺㚱⸛㕡㟡炻㇨ẍ a ᷕ䘬塓攳㕡㔠天⣏㕤ㆾ䫱 㕤暞炻⌛a ≥ 0 ˤ Ȝּ 1ȝ 㯪ᶳ↿⎬㔠䘬⸛㕡㟡烉 (1) 36烊 (2) 16 25烊 (3) 1 2 4 烊 (4) 0.49ˤ

ś

ྋ !! (1) ɻ 2 ( 6)± = 36 ɺ 36 䘬⸛㕡㟡㗗 6± 炻⌛ 36 6 ± = ± (2) ɻ 2 16 4 25 5 = ± ɺ 16 25䘬⸛㕡㟡㗗 4 5 ± 炻⌛ 16 4 25 5 ± = ± (3) ɻ 21 9 4 4 = 炻 2 9 3 4 2 = ± ɺ 21 4 䘬⸛㕡㟡㗗 3 2 ± 炻⌛ 1 9 3 2 4 4 2 ± = ± = ±

(8)

(4) ɻ ( 0.7)± 2 = 0.49 ɺ 0.49 䘬⸛㕡㟡㗗 0.7± 炻⌛ 0.49 0.7 ± = ± Ȝּ 2ȝ ⇌㕟ᶳ↿⎬㔠㚱㰺㚱⸛㕡㟡烉 (1) 64烊 (2) −64烊 (3) 0烊 (4) _{( 4)}− 2 ˤ

ś

ྋ !! (1) ⚈䁢 64 > 0炻㇨ẍ 64 㚱⸛㕡㟡烊 (2) ⚈䁢 64− < 0炻㇨ẍ 64− 㰺㚱⸛㕡㟡烊 (3) 0 㚱⸛㕡㟡烊 (4) ⚈䁢( 4)− 2 =16 > 0炻㇨ẍ_{( 4)}− 2 㚱⸛㕡㟡ˤ

ቚ

!

௫!

1. )αเ*!烉 (1) Ṩ湤㔠䘬⸛㕡䫱㕤 81烎 (2) Ṩ湤㔠䘬⸛㕡䫱㕤 4 9 烎 (3) Ṩ湤㔠䘬⸛㕡䫱㕤 0.25烎 2. 㯪ᶳ↿⎬㔠䘬⸛㕡㟡烉 1ˣ64ˣ1600ˣ0ˣ0.0081ˣ 49 100 ˣ2.25ˣ 25 144 ˤ 3. 㯪ᶳ↿⎬㔠䘬⸛㕡㟡烉 729ˣ324ˣ0.81ˣ0.0225ˣ16 49 ˣ 25 64 4. ⤪⚾炻㯪ⶎ⚰冯⎛⚰ᷕ䘬ˬ烎˭烉 (䫔 4 柴) 8 8 − 3 4 3 4 −

?

1.21 0.36

?

x _x2

(9)

9.2 ზఙπ͞ॲ

ㆹᾹ⶚䴻䞍忻炻ᶨᾳ㬋㔠㚱ℑᾳ⸛㕡㟡炻℞ᷕᶨᾳ㗗㬋㔠炻 ᶨᾳ㗗屈㔠炻᷎ᶼ忁ℑᾳ⸛㕡㟡Ḻ䁢䚠⍵㔠ˤ⚈㬌炻㯪ᶨᾳ㬋㔠䘬⸛㕡㟡炻⎒天㯪↢⬫䘬㬋⸛㕡㟡炻⯙⎗ẍ䞍忻⬫䘬屈⸛㕡㟡ˤ 㬋㔠 a 䘬㬋⸛㕡㟡炻ḇ⎓ a 䘬ᆉೌѳБਥ炻姀 a 炻嬨 ˬ㟡嘇 a˭ˤἳ⤪ 9 䘬䬿埻⸛㕡㟡㗗 3炻⎗ẍ⮓ㆸ 9 = 3炻⍰⤪ 16 = 4ˣ 4 2 25 5 = 䫱ˤ 暞䘬⸛㕡㟡ḇ⎓ 暞䘬䬿埻⸛㕡㟡炻⚈㬌႟ޑᆉೌѳБਥϝ ᙑࢂ႟炻⌛ 0 = 0ˤ Ȝּ 1ȝ 㯪ᶳ↿⎬㔠䘬䬿埻⸛㕡㟡烉 (1) 100烊 (2) 49 64 烊 (3) 0.81ˤ

ś

ྋ !! (1) ɻ 2 10 =100 ɺ 100 䘬䬿埻⸛㕡㟡㗗 10炻⌛ 100 =10 (2) ɻ 2 49 7 64 8 = ɺ 49 64 䘬䬿埻⸛㕡㟡㗗 7 8 炻⌛ 49 7 64 8 = (3) ɻ 0.92 = 0.81 ɺ 0.81 䘬⸛㕡㟡㗗 0.9炻⌛ 0.81 = 0.9

(10)

Ȝּ 2ȝ 㯪ᶳ↿⎬⺷䘬ῤ烉 (1) 10000 烊 (2) − 144 烊 ₍₃₎ 25 121 烊 (4) − 0.0001烊 (5) ± ₆₂₅ 烊 ₍₆₎ 49 81 ± ˤ

ś

ྋ (1) ɻ 2 100 =10000 ɺ 10000 =100 (2) ɻ 122 =144 ɺ − 144 = −12 (3) ɻ 2 25 5 121 11 = ɺ 25 5 121 11 = (4) ɻ (0.01)2 = 0.0001 ɺ − 0.0001 = −0.01 (5) ɻ 252 = 625 ɺ ± 625 = ±25 (6) ɻ 2 49 7 81 9 = ɺ 49 7 81 9 ± = ±

ቚ

!

௫!

1. ⇌㕟ᶳ↿⎬婆⎍⮵ᶵ⮵烉 (1) 5 㗗 25 䘬䬿埻⸛㕡㟡烊 (2) −6㗗 36 䘬䬿埻⸛㕡㟡烊 (3) 6 㗗( 6)− 2 䘬䬿埻⸛㕡㟡烊 (4) 0.4 㗗 0.16 䘬䬿埻⸛㕡㟡

(11)

ቚ

!

(14)

௫ !!

ᗟ

!!

˘!

1. ᶳ朊䘬婆⎍⮵ᶵ⮵烎䁢Ṩ湤烎 (1) −5䘬⸛㕡㗗 25烊 _{(2) 25 䘬⸛㕡㟡㗗 5}− 烊 (3) 49 䘬⸛㕡㟡㗗 7± 烊 ₍₄₎ −₄₉䘬⸛㕡㟡㗗 7− ˤ 2. (1) ⠓堐烉 n 11 12 13 14 15 16 17 18 19 2 n (2) 㟡㒂ᶲ堐炻⮓↢ᶳ↿⎬㔠䘬⸛㕡㟡烉 361ˣ289ˣ196ˣ256ˤ 3. 㯪ᶳ↿⎬㔠䘬⸛㕡㟡冯䬿埻⸛㕡㟡烊 3600ˣ 1 64 ˣ10000ˣ7.29ˣ0.04ˣ 121 289 ˣ 11 1 25 ˤ 4. 㯪ᶳ↿⎬⺷䘬ῤ烉 0 ˣ− 81ˣ 0.09 ˣ _{( 25)}− 2 ˣ 25 36 ± ˤ 5. 㯪ᶳ↿⎬⺷ᷕ䘬 x烉 (1) x =2 25烊 (2) x −2 81= 0烊 (3) 4x =2 49烊 (4) 25x −2 36 = 0ˤ 6. 䓐妰䬿☐㯪ᶳ↿⎬㔠䘬䬿埻⸛㕡㟡 (䱦䡢军⮷㔠⼴䫔Ḵỵ)烉 (1) 10ǹ (2) 13ǹ (3) 97.8ǹ (4) 1.23ǹ (5) 1.491ǹ (6) 14.91ǹ (7) 5.869ǹ (8) 58.69ǹ (9) 867ǹ (10) 7590ǹ (11) 0.759ǹ (12) 0.003094ǹ (13) 87420ǹ (14) 0.46254ǹ (15) 0.00035783Ƕ 7. 䓐妰䬿☐㯪ᶳ↿⎬⺷䘬ῤ (䱦䡢军⮷㔠⼴䫔ᶱỵ)烉 (1) 3.63 ǹ (2) 17.6 ǹ (3) − 2.248 ǹ (4) ± 48.55 ǹ ₍₅₎ _{0.2157 ǹ} ₍₆₎ − _0.09286 ǹ

(15)

(7) 278 3 4 ǹ (8) 6665.72 ± Ƕ

9.4 ϲ͞ॲ

ㆹᾹἮ䚳ᶳ朊䘬⓷柴烉天 ᶨ⎒㬋㕡橼䘬㛐䭙炻ἧ⬫䘬⭡䧵㗗 125 m3炻忁ᾳ㛐䭙䘬䧄攟ㅱ䔞㗗⣂⮹ m烎 ⚈䁢㬋㕡橼䘬⭡䧵䫱㕤䧄攟ᷳ䩳㕡炻⤪㝄姕䧄攟䁢 x m炻㟡 㒂柴シ炻⼿ 3 125 x = ˤ 忁⯙㗗天㯪↢ᶨᾳ㔠炻ἧ⬫䘬䩳㕡䫱㕤 125ˤ ⚈䁢53 =125炻㇨ẍ炻忁ᾳ㬋㕡橼㛐䭙䘬䧄攟㗗 5 mˤ ᶨ凔⛘炻⤪㝄ᶨᾳ㔠䘬䩳㕡䫱㕤 a炻忁ᾳ㔠⯙⎓ a 䘬ҥБ ਥ(ḇ⎓ ΟԛБਥ)ˤ㎃⎍娙婒炻⤪㝄 3 x = a 炻恋湤 x ⯙⎓ a 䘬䩳㕡㟡ˤ㔠 a 䘬䩳㕡㟡䓐䫎嘇ˬ 3 _{˭堐䣢炻嬨ἄˬᶱ㫉㟡嘇 a˭}_炻 ℞ᷕ a 㗗塓攳㕡㔠炻3 㗗㟡㊯㔠ˤἳ⤪ 3 5 =125炻5 ⯙㗗 125 䘬䩳㕡㟡炻䓐⺷⫸堐⺷⯙㗗 3 125 = 5ˤ 㯪ᶨᾳ㔠䘬䩳㕡㟡ᷳ忳䬿炻⎓ ໒ҥБˤ攳䩳㕡冯䩳㕡Ḻ䁢微忳䬿ˤ Ȝּ 1ȝ 㯪ᶳ↿⎬㔠䘬䩳㕡㟡烉 (1) −8烊 _{(2) 8烊} ₍₃₎ 8 27 − 烊 (4) 0.216烊 (5) 0ˤ

ś

ྋ !! (1) ɻ 3 ( 2)− = −8 ɺ −8䘬䩳㕡㟡㗗 2− 炻⌛ 3 8 2 − = −

(16)

(2) ɻ 23 = 8 ɺ 8 䘬䩳㕡㟡㗗 2炻⌛ 3 8 = 2 (3) ɻ 3 8 2 27 3 = − − ɺ 8 27 − 䘬䩳㕡㟡㗗 2 3 − 炻⌛ 3 8 2 27 3 − = − (4) ɻ 0.63 = 0.216 ɺ 0.216 䘬䩳㕡㟡㗗 0.6炻⌛ 3 0.216 = 0.6 (5) ɻ 03 = 0 ɺ 0 䘬䩳㕡㟡㗗 0炻⌛ 3 0 = 0 ⽆ἳ 1 ⎗ẍ䚳↢炻㬋㔠 8 冯 0.216 ↮⇍㚱ᶨᾳ㬋䘬䩳㕡㟡 2 冯 0.6烊屈㔠 8− 冯 8 27 − ↮⇍㚱ᶨᾳ屈䘬䩳㕡㟡 2− 冯 2 3 − 烊暞㚱䩳㕡㟡暞ˤᶨ凔⛘炻҅ኧԖ΋ঁ҅ޑҥБਥǹॄኧԖ΋ঁॄޑҥБ ਥǹ႟ޑҥБਥϝᙑࢂ႟ˤ ⽆ἳ 1 怬⎗ẍ䚳↢炻 ɻ 3 −8 = −2 3 8 = 2 ɺ 3 3 8 8 − = − ᶨ凔⛘炻⤪㝄a > 炻恋湤0 3 −a = −3 aˤ㇨ẍ炻㯪屈㔠䘬䩳㕡㟡炻⎒天⃰㯪↢忁ᾳ屈㔠䘬䳽⮵ῤᷳ䩳㕡㟡炻䃞⼴ℵ⍾⬫䘬䚠⍵ 㔠ˤ

(17)

ቚ

!

௫!

1. (1) ⮓↢ 1ˣ2ˣ3ˣ4ˣ5ˣ6ˣ7ˣ8ˣ9 忁ḅᾳ㔠䘬䩳㕡烊 (2) ⇑䓐䫔(1)⮷柴䘬䳸㝄炻㯪ᶳ↿⎬㔠䘬䩳㕡㟡烉 27ˣ 64− ˣ343ˣ 729− ˤ 2. 㯪ᶳ↿⎬㔠䘬䩳㕡㟡烉 1ˣ512ˣ 8 227 ˣ 27 64 − ˣ1 8 ˣ 1 8 − ˣ0.125ˣ ₁₅5 8 − ˤ Ȝּ 2ȝ 㯪ᶳ↿⎬⺷䘬ῤ烉 (1) 3 27 烊 (2) 3 −27 烊 (3) 3 210 27 − 烊 ₍₄₎ 3 27 64 − − ˤ

ś

ྋ !! (1) 3 27 = 3烊 (2) 3 −27 = −3 27 = −3烊 (3) 3 210 3 64 4 11 27 27 3 3 − = − = − = − 烊 (4) 3 27 3 27 3 64 64 4 − − = = ˤ

ቚ

!

௫!

1. 㯪ᶳ↿⎬⺷䘬ῤ烉 3 1000 ˣ−3 0.001 ˣ 3 64 125 − ˣ−3 −₂₁₆ ˣ 3 −₁ˣ 3 33 8 ˤ 2. ⠓⮓ᶳ堐烉 a 0.000001 0.001 1 1000 1000000 3 a 奨⮇ᶲ堐᷎婒㖶䔞⶚䞍㔠 a 䘬⮷㔠溆⎹⎛(ㆾ⎹ⶎ)㭷䦣≽ᶱ ỵ炻⬫䘬䩳㕡㟡 3 a ᷳ⮷㔠溆䘬䦣≽夷⼳㗗⾶㧋䘬烎

(18)

ቚ

!

௫!

8

烊 (9) −80烊 ₍₁₀₎ −_16.5ˤ ⮷㔠溆⎹⎛ 䦣≽ᶱỵ ⮷㔠溆⎹⎛ 䦣≽ᶨỵ 3 0.00525 ≈ 0.1738 ⮷㔠溆⎹⎛ 䦣≽ᶱỵ ⮷㔠溆⎹⎛ 䦣≽ᶨỵ ⮷㔠溆⎹⎛ 䦣≽ᶱỵ 3 5.25 ≈ 1.738 3 5250 ≈ 17.38 3 5250000 ≈ 173.8 ⮷㔠溆⎹⎛ 䦣≽ᶨỵ

(21)

ቚ

!

௫!

2. 䓐妰䬿☐㯪ᶳ↿⎬⺷䘬ῤ (䱦䡢军⮷㔠⼴䫔Ḵỵ)烉 (1) 3 5 烊 (2) 3 68.8 烊 (3) 3 0.8725 烊 (4) 3 −43.6 烊 ₍₅₎ −3 _0.459 烊 ₍₆₎ 3 _{0.00094 烊} (7) 3 0.037 烊 (8) 3 6570 烊 (9) 3 420000 烊 (10) 3 0.00005087 烊 (11) 3 −258 烊 ₍₁₂₎ −3 _0.0062548 3. 㯪ᶳ↿⎬⺷ᷕ䘬 x烉 (1) x =3 27 烊 (2) 3 1 8 x = 烊 (3) 27x +3 64 = 0烊 ₍₄₎ ₄_{x − =}3 ₅ ₀ _{(䱦䡢⇘ 0.01)ˤ}

9.6 ၁ᇴ

ㆹᾹ⶚䴻⬠忶炻ảỽᶨᾳ㚱䎮㔠悥⎗ẍ⮓ㆸ㚱旸⮷㔠(㔜㔠⎗ ẍ䚳 ⮷㔠溆⼴朊㗗 0 䘬⮷㔠)ㆾ侭⽒䑘⮷㔠䘬⼊⺷ˤἳ⤪烉 3 3.0 3 0.6 5 9 0.81 11 = − = − = ⍵忶Ἦḇ⮵炻⌛ảỽ㚱旸⮷㔠冯⽒䑘⮷㔠悥㗗㚱䎮㔠ˤ ⛐⮎晃ᷕ炻ㆹᾹ怬㚫忯⇘ᶨ䧖⮷㔠炻⬫Ᾱᶵ傥⮓ㆸ㚱旸⮷㔠䘬⼊⺷炻ḇᶵ傥⮓ㆸ⽒䑘⮷㔠䘬⼊⺷ˤἳ⤪炻 2 1.41421356= ˣ 0.101001000100001 (ℑᾳ 1 ᷳ攻ὅ㫉⣂ᶨᾳ 0)ˣ⚻␐䌯π = 3.14159265 炻⬫Ᾱ䘬⮷㔠ỵ㔠㗗䃉旸䘬炻侴ᶼ㗗ᶵ⽒䑘䘬ˤ忁㧋䘬⮷㔠⎓ 䃉旸ᶵ⽒䑘⮷㔠ˤ 䃉旸 ᶵ ⽒䑘 ⮷ 㔠 ⯙ ⎓ ค ౛ ኧ ˤ ⤪ ᶲ 朊 ἳ ⫸ ᷕ 䘬 2 ˣ ˣ 悥㗗䃉䎮㔠ˤ䃉䎮㔠㗗⼰⣂䘬炻ἳ⤪炻

(22)

3 3 1.732050 5 2.236067 2 1.259921 1.57079632 2 0.232332333233332 ( 2 3) π = − = − = = ℑᾳ ᷳ攻ὅ㫉⣂ᶨᾳ ḇ悥㗗䃉䎮㔠ˤỮ㗗⁷ 4ˣ 3 27 − 䫱⯙ᶵ㗗䃉䎮㔠炻侴㗗㚱䎮㔠ˤ 䃉䎮㔠⎗↮䁢㬋䃉䎮㔠冯屈䃉䎮㔠ˤἳ⤪ 2 ˣ 5 3 ˣπ ˣɃ 㗗㬋䃉䎮㔠烊− 2 ˣ−5 ₃ ˣ−π ˣɃ㗗屈䃉䎮㔠ˤ 㚱䎮㔠冯䃉䎮㔠䴙䧙ჴኧˤ忁㧋炻ㆹᾹ⬠忶䘬㔠⯙㚱烉㬋㚱䎮㔠㚱䎮㔠暞㚱旸⮷㔠ㆾ⽒䑘⮷㔠屈㚱䎮㔠 ⮎㔠㬋䃉䎮㔠䃉䎮㔠䃉旸ᶵ⽒䑘⮷㔠屈䃉䎮㔠㚱䎮㔠冯䃉䎮㔠悥㚱㬋屈ᷳ↮炻㇨ẍ⮎㔠ḇ㚱㬋屈ᷳ↮ˤ⤪ 㝄 a 堐䣢ᶨᾳ㬋⮎㔠炻 a− ⯙堐䣢ᶨᾳ屈⮎㔠ˤa 冯 a− Ḻ䁢䚠⍵ 㔠ˤㆹᾹ夷⭂烉0 䘬䚠⍵㔠ṵ冲㗗 0ˤ ⮎㔠䘬䳽⮵ῤシ佑ḇ冯㚱䎮㔠ᶨ㧋烉΋ঁ҅ჴኧޑ๊ჹॶࢂ Ѭҁيǹ΋ঁॄჴኧޑ๊ჹॶࢂѬޑ࣬ϸኧǹ႟ޑ๊ჹॶࢂ႟Ƕ ἳ⤪炻| 2 |= 2 ˣ| − _{2 |}= ₂ ˣ| |π =π ˣ|−π _|= π ˣ| 0 | 0= 䫱ˤ ㆹᾹ⶚䴻䞍忻炻㭷ᶨᾳ㚱䎮㔠悥⎗ẍ䓐㔠庠ᶲ䘬溆Ἦ堐䣢ˤ Ữ㗗炻㔠庠ᶲ㇨㚱䘬溆᷎ᶵ悥㗗堐䣢㚱䎮㔠ˤ⚈䁢㔠庠ᶲ䘬溆⎗ ẍ堐䣢ảỽ㔜㔠ㆾ⮷㔠炻℞ᷕ䘬䃉旸ᶵ⽒䑘⮷㔠⯙㗗䃉䎮㔠ˤ ㈲㔠⽆㚱䎮㔠㒜⃭⇘⮎㔠ẍ⼴炻⮎㔠冯㔠庠ᶲ䘬溆⯙㗗ᶨᶨ ⮵ㅱ䘬炻⌛㭷ᶨᾳ⮎㔠悥⎗ẍ䓐㔠庠ᶲ䘬ᶨᾳ溆Ἦ堐䣢炻⍵忶 Ἦ炻㔠庠ᶲ䘬㭷ᶨᾳ溆悥堐䣢ᶨᾳ⮎㔠ˤ

(23)

⛐忚埴⮎㔠忳䬿㗪炻㚱䎮㔠䘬忳䬿⼳冯忳䬿⿏岒⎴㧋怑䓐ˤ ⛐⮎㔠䭬⚵ℏ炻㬋㔠冯暞䷥⎗ẍ忚埴攳㕡忳䬿炻枰天㲐シ炻屈㔠 ⎒傥攳⣯㫉㕡炻ᶵ傥攳„㫉㕡ˤ ⛐⮎㔠忳䬿ᷕ炻⤪㝄忯⇘䃉䎮㔠炻᷎ᶼ暨天㯪↢䳸㝄䘬役Ụ ῤ炻⎗ẍ㊱䄏㇨天㯪䘬䱦䡢⹎䓐役Ụ䘬㚱旸⮷㔠⍣ẋ㚧䃉䎮㔠炻 ℵ忚埴妰䬿ˤ Ȝּȝ 妰䬿烉 (1) 5 +π (䱦䡢⇘ 0.01)烊 (2) 3 2 (䳸㝄ᾅ䔁ᶱᾳ㚱㓰㔠⫿)ˤ

ś

ྋ !!! (1) 5 +π ≈ 2.236 3.142+ ≈ 5.38烊 (2) 3 2 ≈1.732 1.414× ≈ 2.45ˤ

ቚ

!

௫!

1. )αเ*!Ṩ湤⎓ 䃉䎮㔠烎冱↢ℑᾳ䃉䎮㔠䘬ἳ⫸ˤ 2. )αเ*!ᶳ朊䘬婆⎍⮵ᶵ⮵烎⤪㝄ᶵ⮵炻⾶㧋婒ㇵ⮵烎 (1) 䃉旸⮷㔠悥㗗䃉䎮㔠烊 (2) 䃉䎮㔠悥㗗䃉旸⮷㔠烊 (3) ⷞ㟡嘇䘬㔠悥㗗䃉䎮㔠ˤ 3. ᶳ↿⎬㔠炻⒒ṃ㗗㚱䎮㔠炻⒒ṃ㗗䃉䎮㔠烎 π − ˣ 3.14− ˣ− ₃ ˣ1.732ˣ0ˣ0.3ˣ 18ˣ 25 36 ˣ 21 31 ˣ 7 ˣ 16 − ˤ 4. (1) 㚱䎮㔠悥㗗⮎㔠╶烎⮎㔠悥㗗㚱䎮㔠╶烎冱ἳ婒㖶烊 (2) 䃉䎮㔠悥㗗⮎㔠╶烎⮎㔠悥㗗䃉䎮㔠╶烎冱ἳ婒㖶ˤ 5. 㯪ᶳ↿⎬⺷ᷕ䘬⮎㔠 x烉 (1) | | 2 3 x = 烊 (2) | | 0x = ǹ (3) | |x = 10 烊 (4) | |x = π ˤ

(24)

ቚ

!

௫!

6. 妰䬿烉 (1) 10 6 7 π + − _{(䱦䡢⇘ 0.01)烊} (2) ( 4) 7 1 6 2 − × + × _{(䳸㝄ᾅ䔁ᶱᾳ㚱㓰㔠⫿)ˤ}

௫ !!

ᗟ

!!

˟!

1. ᶳ↿⎬婆⎍⮵ᶵ⮵烎䁢Ṩ湤烎 (1) −0.064䘬䩳㕡㟡㗗 0.4− 烊 (2) 8 䘬䩳㕡㟡㗗 2± 烊 (3) 1 27 䘬䩳㕡㟡㗗 1 3 ˤ 2. 㯪ᶳ↿⎬⺷䘬ῤ烉 (1) −3 0.027 ǹ ₍₂₎ 3 −₃₄₃ ǹ ₍₃₎ 3 125 27 ǹ (4) 3 1 61 64 + ǹ ₍₅₎ 3 1 19 27 − ǹ ₍₆₎ 3 7 1 8 − − Ƕ 3. 㯪ᶳ↿⎬⺷ᷕ䘬 x烉 (1) x =3 0.729烊 (2) 64x +3 125 = 0烊 (3) 3 3 3 8 x − = 烊 (4) (x −1)3 =8ˤ 4. 䓐妰䬿☐㯪ᶳ↿⎬㔠䘬䩳㕡㟡 (䱦䡢军⮷㔠⼴䫔Ḵỵ)烉 (1) 0.39ǹ (2) 48.3ǹ (3) −2.36ǹ (4) −34.26ǹ _{(5) 434.5ǹ} _{(6) 4936ǹ} (7) −0.0532ǹ _{(8) 0.007283Ƕ}

(25)

5. 䓐妰䬿☐㯪ᶳ↿⎬⺷䘬ῤ (䱦䡢军⮷㔠⼴䫔Ḵỵ)烉 (1) 3 0.432 ǹ (2) 3 −1.948 ǹ ₍₃₎ −3 _7456.3ǹ (4) 3 67.5 ǹ (5) 3 400000 ǹ (6) 3 85 9 − ǹ (7) 3 0.0000518ǹ (8) 3 350 4 25 − − Ƕ 6. (1) 㬋㕡橼䘬堐朊䧵㗗 11 m2炻㯪⬫䘬䧄攟 (䱦䡢⇘ 0.01 m)烊 (2) 㬋㕡橼䘬橼䧵㗗 11 m2炻㯪⬫䘬堐朊䧵 (䱦䡢⇘ 0.01 m)ˤ 7. ⶚䞍a =0.5ˣb = 7.5ˣc = −0.36炻妰䬿ᶳ↿⎬⺷䘬ῤ(䱦䡢⇘ 0.01)烉 (1) 2 4 2 b b ac a − ± − 烊 (2) (3 b − 3 a c) ˤ 8. ㈲ᶳ↿⎬㔠↮⇍⠓⛐㚱䎮㔠普⎰冯䃉䎮㔠普⎰䘬⚰墉烉 22 7 ˣ3.14159265ˣ 8 ˣ 8 − ˣ 3 _{9 ˣ} 0.6ˣ31 4 ˣ 36 ˣ1.732ˣ 3 π ˤ (䫔 8 柴) 9. ᶳ↿⎬婆⎍⮵ᶵ⮵烎䁢Ṩ湤烎 (1) ㇨㚱䘬㚱䎮㔠悥⎗ẍ䓐㔠庠ᶲ䘬溆堐䣢↢Ἦ烊⍵忶Ἦ炻㔠庠ᶲ䘬溆悥堐䣢㚱䎮㔠烊 (2) ㇨㚱䘬⮎㔠悥⎗ẍ䓐㔠庠ᶲ䘬溆堐䣢↢Ἦ烊⍵忶Ἦ炻㔠庠ᶲ䘬溆悥堐䣢⮎㔠ˤ 㚱䎮㔠普⎰ 䃉䎮㔠普⎰

(26)

10. 㯪ᶳ↿⎬㔠䘬䳽⮵ῤ烉 3 − ˣ 3 −₈ ˣ 7 ˣ 2 3 − ˣ 3 1.7 − ˣ1.4− ₂ ˤ 11. 㭼庫ᶳ↿⎬䳬墉ℑᾳ㔠䘬⣏⮷烉烉 (1) 15ˣ0烊 (2) 3 4 − ˣ0烊 (3) −7ˣ 9− 烊 ₍₄₎ _{2 ˣ}− ₃ 烊 (5) − 3 ˣ 1.731− 烊 ₍₆₎ π ˣ3.1416 12. 妰䬿 (䱦䡢⇘ 0.01)烉 (1) 5 − 3 +0.415烊 ₍₂₎ 3 ₆ −π − ₂ ˤ

!

̈

!!

ඕ!

!

ᶨˣ㛔䪈ᷣ天ℏ⭡㗗㔠䘬攳㕡ᷳ㤪⾝ẍ⍲⮎㔠䘬⇅㬍㤪⾝ˤ Ḵˣ㔠䘬㤪⾝㗗⛐⮎嶸ᷕᶵ㕟䘤⯽䘬ˤ⛐⮷⬠墉ㆹᾹ⃰⬠佺 Ḯ冒䃞㔠冯暞烊⍰⬠佺Ḯ↮㔠冯⮷㔠(㚱旸⮷㔠冯⽒䑘⮷㔠)炻⯙ 㗗婒炻⬠佺Ḯ㬋㚱䎮㔠冯暞烊⇘Ḯᶫ⸜䳂⍰⬠佺Ḯ屈㚱䎮㔠ˤ忁㧋炻㔠䘬㤪⾝⯙㒜⃭⇘Ḯ㚱䎮㔠ˤ䎦⛐ㆹᾹ⍰㈲㔠䘬㤪⾝㒜⃭⇘ ⮎㔠ˤ忁㧋炻ㆹᾹ⬠忶䘬㔠⎗ẍ↿ㆸᶳ堐烉㬋㚱䎮㔠㚱䎮㔠暞屈㚱䎮㔠 ⮎㔠㬋䃉䎮㔠䃉䎮㔠屈䃉䎮㔠 ᶱˣ⛐忚埴⮎㔠忳䬿㗪炻㚱䎮㔠䘬忳䬿⼳冯⿏岒⎴㧋怑䓐ˤ

(27)

攳㕡冯Ḁ㕡Ḻ䁢微忳䬿ˤ䓐Ḁ㕡⎗㩊槿攳㕡䘬䳸㝄㬋䡢⏎ˤ ⛐⮎㔠䭬⚵ℏ炻㬋㔠㚱 n 㫉㕡㟡烊屈㔠㚱⣯㫉㕡㟡炻Ữ㰺㚱 „㔠㕡㟡烊暞䘬 n 㫉㕡㟡㗗暞ˤ 㬋㔠ᷳ㬋䘬㕡㟡⎓ 䬿埻㟡炻暞䘬䬿埻㟡㗗暞ˤ

ኑ௫ણ҂ᗟ˝!

1. 妰䬿烉 (1) 8 1 1 0.25 2 21 6 7 6 3 3 − − − + − − × ÷ × 烊 (2) 2 2 2 4 2 2 4 4 3 5 3 3 5 5 − − − + × − + − 烊 (3) 7 ( 3)3 1 ( 3)2 1 ( 3) 1 9 6 12 4 × − − × − + × − − 烊 (4) 8.9543 −7.3083 − 26.542 +37.692_{(䓐妰䬿☐妰䬿炻䱦䡢⇘} 0.1)ˤ 2. ⎬冱ᶨἳ婒㖶≈㱽Ṍ㎃⼳ˣ≈㱽䳸⎰⼳ˣḀ㱽Ṍ㎃⼳ˣḀ㱽䳸⎰⼳冯↮惵⼳ˤ 3. ⤪㝄a ≠ 炻恋湤0 a 䘬䚠⍵㔠㗗Ṩ湤烎a 䘬Ὰ㔠㗗Ṩ湤烎a 䘬䳽 ⮵ῤ㗗Ṩ湤烎 4. 冱ἳ婒㖶烉 (1) a 㗗ᶵ㗗ᶨ⭂⣏㕤−a烊 (2) 2 a 㗗ᶵ㗗ᶨ⭂⮷㕤 a烊 (3) ⶚䞍a2 = b2炻傥ᶵ傥㕟⭂ a b= 烊 (4) ⶚䞍| | | |a > b 炻傥ᶵ傥㕟⭂ a b> ˤ 5. ⤪㝄 n 㗗㬋㔜㔠炻恋湤 (1) 2n 㗗⣯㔠怬㗗„㔠烎 2n − ␊烎1 (2) ( 1)− 2n =烎_{( 1)}− 2n−1 = 烎₍−_a₎2n =烎₍−_a₎2n−1 = 烎

(28)

6. ⶚䞍 A = x3 − x +6ˣ_B = _x3 − _x2 + _x+₃炻妰䬿烉 (1) (A x− 3)(B− x3)烊 _{(2) (}_A+ _B_{) (}÷ _{A B}− ₎烊 (3) (A B− )2 ˤ 7. ㈲ᶳ↿⎬⺷↮妋⚈⺷烉 (1) m3 +12m n2 +36mn2烊 ₍₂₎ ₁₂_xy −₉_x2 −₄_y2烊 (3) x4 −13x2 +36 烊 ₍₄₎ _a6 −₇_a3 −₈烊 (5) a3 −b3 −3ab a b( − )烊 ₍₆₎ _a3 +_b3 +₃_{a b}2 +₃_ab3烊 (7) a3 −b3 − a2 +b2烊 ₍₈₎ _{a a}₍ +₁₎−_{b b}₍ +₁₎烊 (9) a2 −4ab+ 4b2 −4c2烊 (10) (x + z x)( − z)+ y y( −2 )x 烊 (11) (5x2 + x− 20)2 −(4x2 − x+ 4)2烊 (12) (a b+ )2 +(a+c)2 −(c+ d)2 −(b d+ )2ˤ 8. ⊾䯉烉 (1) (x −1)(x+ 4) {(− x +2)(x −5) [(− x+3)(x −4) (− x−1)(x+ 6)]} (2) (2x2 −6x+5)2 −(2x2 −6x+4)2 −(2x+3)2烊 (3) 3 3 3 3 2 a b a b ab a b a b − + − − − + 烊 (4) 2 2 2 2 2 2 3 2 12 5 4 9 4 4 5 6 x x x x x x x x x x x − + − − − + ÷ − − + − + 烊 (5) ₂ 2 ₂ 2 ₂ 2 ₂1 3 2 4 3 5 6 1 x x x x x x x − + − − + − + − + − 烊 (6) 2 3 2 2 2 2 2 2 a a a a a b a ab b a b a b ÷ − − + + + + − 烊 (7) 1 ₁ 1 1 x _y y _x x y y x + ₊ ÷ − − 烊 (8) 1 1 1 1 1 x − − ˤ

(29)

9. (1) ⶚䞍 A= 2x4 −3x2 +6x −7ˣ_B = _x2 − _x +₁炻㯪 A B÷ 䘬⓮ ⺷ Q ⍲检⺷ R烊 (2) 㟡㒂 A= BQ+ RἮ㩊槿䫔(1)⮷柴ᷕ㇨⼿䘬䳸㝄ˤ 10. (1) 㕡䦳㚱⒒⸦㡅⎴妋⍇䎮烎 (2) ᶵ䫱⺷㚱⒒⸦㡅➢㛔⿏岒烎 11. 妋ᶳ↿㕡䦳烉 (1) 3 3 1 1 2 3 5 2 4 5 x x x− − − − x− + = x− 烊 (2) 1 ( 1)2 5( 2)2 1( 3)2 0 2 x 6 x 3 x − − + + − = 烊 (3) 3 ₂ 38 4 4 4 4 1 1 x x x x − − = − + − 烊 (4) 5 5 100 ₂ 5 5 25 x x x x x + − + = − + − ˤ 12. 妋ᶳ↿㕡䦳䳬烉 (1) 7 3 5 5 3 2 3( 1) 5( 1) x y x y x y x y − − + = − − = + 烊 (2) 3 2 2 3 1 6 7 3 2 2 3 5 6 7 x y x y x y x y − + + = − + − = 烊 (3) 11 18 13 2 3 3 2 27 2 1 3 2 2 3 x y x y x y x y + = − − − = − − 烊 (4) 3 14 4 6 7 1 2 8 5 18 2 3 x y z y z x z x y + = + − = + + = + ˤ

(30)

13. 妋ᶳ↿㕡䦳䳬烉 (1) 2 2 2 2 ax by a b bx ay a b − = + + = + 烊 (2) 2 4 x y a a b a b x y ab + = + − − = 烊 (3) a b a b x y a b a b x y + = − − = + 烊 (4) 2 2 2 y z a z x b x y c + = + = + = ˤ 14. ⶚䞍㕡䦳x3 + px2 +qx+ r = 0 㚱ᶨᾳ㟡㗗 1炻ᶨᾳ㟡㗗 2炻ᶨᾳ 㟡㗗 3炻㯪 pˣqˣr 䘬ῤˤ 15. (1) x ⍾Ṩ湤ῤ㗪炻ᶵ䫱⺷16 3(2− x −5) < 25 4− xㆸ䩳烎 (2) x ⍾Ṩ湤ῤ㗪炻ẋ㔠⺷ 5(x −1) 6(− x −2)䁢㬋㔠烎 (3) x ⍾Ṩ湤ῤ㗪炻ẋ㔠⺷1( 2) 1( 1) 5 x 3 x + − − 䁢屈㔠烎 (4) 㯪ᶵ䫱⺷ 4− < x ≤ −1䘬㔜㔠妋炻᷎⛐㔠庠ᶲ堐䣢↢Ἦ烊 (5) 㯪ᶵ䫱⺷| | 3x < 䘬㔜㔠妋炻᷎⛐㔠庠ᶲ堐䣢↢Ἦˤ 16. (1) ⤪㝄 a b≠ 炻恋湤₍_{a b}− ₎2 > ₀ 炻䁢Ṩ湤烎 (2) ⤪㝄 a ≠ b炻恋湤_a2 +_b2 > ₂_ab炻䁢Ṩ湤烎 17. ㈲ᶨᾳ攟㕡⼊䘬攟⡆≈ 4 cm炻⮔㷃⮹ 1 cm炻⬫䘬朊䧵ᶵ嬲烊㈲⬫䘬攟㷃⮹ 2 cm炻⮔⡆≈ 1 cm炻⬫䘬朊䧵ḇᶵ嬲ˤ㯪忁ᾳ 攟㕡⼊䘬朊䧵ˤ 18. 䓚ˣḁℑ昲ᾖᶨ㡅嶗炻⤪㝄ℑ昲⎰ᾖ 4 ⣑⼴炻∑检悐↮䓙䓚昲╖䌐ᾖ炻⃰⼴ℙ天 6 ⣑烊⤪㝄ℑ昲⎰ᾖ 3 ⣑⼴炻∑检悐↮ 䓙ḁ昲╖䌐ᾖ炻⃰⼴ḇℙ天 6 ⣑ˤ㯪ℑ昲╖䌐ᾖ忁㡅嶗炻⎬ 暨天⸦⣑ˤ 19. (1) 姕x2 = a炻a 㗗 x 䘬Ṩ湤㔠烎x 㗗 a 䘬Ṩ湤㔠烎 (2) 姕 y3 = b炻b 㗗 y 䘬Ṩ湤㔠烎y 㗗 b 䘬Ṩ湤㔠烎

(31)

20. ᶳ↿婆⎍⮵ᶵ⮵烎䁢Ṩ湤烎 (1) −4䘬⸛㕡㗗 16烊 _{(2) 16 䘬⸛㕡㟡㗗 4}− 烊 (3) −1䘬䩳㕡㗗 1− 烊 ₍₄₎ −₁䘬䩳㕡㟡㗗 1− 烊 (5) −1䘬⸛㕡㟡㗗 1− 烊 _{(6) 0 䘬⸛㕡㟡㗗 0ˤ} 21. Ṩ湤⎓䬿埻⸛㕡㟡烎㯪↢ᶳ↿⎬㔠䘬⸛㕡㟡烉 64ˣ36ˣ0.25ˣ5ˣ( 3)− 2ˣ 2 2 5 ˣ 2 4 13 − ˤ 22. 䓐妰䬿☐㯪ᶳ↿⎬⺷䘬ῤ烉 (1) 94.3 ǹ (2) 0.5374 ǹ (3) 0.07321 ǹ (4) 8200 ǹ (5) 527.48 ǹ (6) 41 4 ǹ (7) 3 35.8 ǹ (8) 3 0.43 ǹ (9) 3 278.4 ǹ (10) 31.47 ǹ (11) 3 0.002008ǹ (12) 3 4107 1 2 ǹ (13) 0.00895 − 0.0234 ǹ (14) 3 _0.0695 + 3 _0.1783 ǹ (15) 3.4722 +5.0892 ǹ ₍₁₆₎ 2 2 3.472 +5.089 Ƕ 23. 㯪ᶳ↿⎬⺷ᷕ䘬 x烉 (1) x =2 169ǹ (2) 121x −2 25 = 0 (3) 9x =2 64ǹ (4) x −2 1.69 = 0ǹ (5) _{x =}3 ₆₄₀₀₀ǹ ₍₆₎ _{x = −}3 _0.125ǹ 24. ⶚䞍㬋㕡⼊䘬朊䧵㗗 360 cm2炻㯪㬋㕡⼊怲攟(䱦䡢⇘ 0.1 cm)ˤ 25. ⶚䞍㬋㕡橼䘬䧄攟㗗 5 cm炻ℵἄᶨᾳ㬋㕡橼炻ἧ⬫䘬橼䧵㗗 ⍇㬋㕡橼䘬橼䧵ᷳ 2 ᾵炻㯪㇨䘬㬋㕡橼ᷳ䧄攟 (䱦䡢⇘ 0.1 cm)ˤ 26. 天ἧṢ忈⛘䎫堃㗇傥⣈丆⛘䎫忳廱炻⽭枰ἧ⬫䘬忇⹎⣏㕤䫔 ᶨ⬯⭁忇⹎侴⮷㕤䫔Ḵ⬯⭁忇⹎ˤ䫔ᶨ⬯⭁忇⹎䘬妰䬿℔⺷㗗V₁ = qR _{(m/䥺)炻䫔Ḵ⬯⭁忇⹎䘬妰䬿℔⺷㗗} 2 2 V = qR (m/䥺)炻R = 6.4 10× 6 _{(m)ˤ㯪䫔ᶨˣ䫔Ḵ⬯⭁忇⹎(䱦䡢⇘ 100} m/䥺)ˤ

(32)

27. Ṩ湤⎓ 䃉䎮㔠烎⛐ᶳ朊㭷ᶨᾳ⚰墉炻军⮹⠓ℍᶱᾳ㔠烉 28. 㚱㰺㚱㚨⮷䘬冒䃞㔠烎㚱㰺㚱㚨⮷䘬㔜㔠烎㚱㰺㚱㚨⮷䘬㚱䎮㔠烎㚱㰺㚱㚨⮷䘬䃉䎮㔠烎㚱㰺㚱㚨⮷䘬⮎㔠烎㚱㰺㚱䳽 ⮵ῤ㚨⮷䘬⮎㔠烎 29. 㯪ᶳ↿⎬㔠䘬䳽⮵ῤ烉 2 5 − ˣ 3 −₇ ˣ 5 − ₇ ˣ 3.1416−π ˤ 30. 㭼庫ᶳ↿⎬䳬墉ℑᾳ⮎㔠䘬⣏⮷烉烉 (1) 1.574ˣ1.5烊 (2) − 5 ˣ 2.24− 烊 (3) −π ˣ 3.1415926− 烊 ₍₄₎ _{29 ˣ}₅ 4 13 ˤ 31. 妰䬿(䱦䡢⇘ 0.01)烉 (1) 10 1 0.145 3 π + − + 烊 ₍₂₎ ₅ 1 _4.375 4 7 3 + − − ˤ 32. 奨⮇⸛㕡㟡堐炻⚆䫼ᶳ↿⓷柴烉 (1) 塓攳㕡㔠䓙 1.0 ⡆≈⇘ 1.1 㗪炻⸛㕡㟡䘬ῤ㗗⡆⣏怬㗗㷃⮷烎 (2) 塓攳㕡㔠䘬ῤ徸㻠⡆⣏㗪炻⸛㕡㟡䘬ῤ㗗⡆⣏怬㗗㷃 ⮷烎㬋㚱䎮㔠普⎰ 屈㚱䎮㔠普⎰ 㬋䃉䎮㔠普⎰ 屈䃉䎮㔠普⎰ (䫔 27 柴)

(33)

ܢᐂ!

π͞ॲ۞ඊზՐڱ!

㯪ᶨᾳ㬋㔠䘬⸛㕡㟡炻昌㞍⸛㕡㟡堐⢾炻怬⎗ẍ䓐䫮䬿Ἦ㯪ˤ ㈲ᶨᾳ㬋㔜㔠攳⸛㕡炻椾⃰天䡢⭂⸛㕡㟡㚱⸦ỵ㔜㔠ˤ ㆹᾹ䞍忻炻ℑᾳ㬋㔠墉炻庫⣏䘬ᶨᾳ㬋㔠ᷳ⸛㕡ḇ庫⣏ˤ䓙 ⸛㕡忳䬿⼿⇘炻 2 2 2 2 2 2 1 1 9 81 10 100 99 9801 100 10000 999 998001 = = = = = = ⎗ẍ䚳↢炻ᶨỵ㔠䘬⸛㕡㗗ᶨỵ㔠ㆾ侭Ḵỵ㔠炻Ḵỵ㔠䘬⸛㕡㗗 ᶱỵ㔠ㆾ侭⚃ỵ㔠烊ᶱỵ㔠䘬⸛㕡㗗Ḽỵ㔠ㆾ侭ℕỵ㔠炻Ƀ烊⍵ 忶Ἦ炻 ᶨỵ㔠ㆾ侭Ḵỵ㔠䘬⸛㕡㟡㚱ᶨỵ㔜㔠炻 ᶱỵ㔠ㆾ侭⚃ỵ㔠䘬⸛㕡㟡㚱ℑỵ㔜㔠炻 Ḽỵ㔠ㆾ侭ℕỵ㔠䘬⸛㕡㟡㚱ᶱỵ㔜㔠炻 ⚈㬌炻㈲塓攳⸛㕡㔠⽆⎛⇘ⶎ㭷ℑỵ䁢ᶨ㭝䓐ᶨᾳ㐯嘇ˬ’˭ ↮攳(㚨ⶎ怲䘬ᶨ㭝⎗ẍ⎒㚱ᶨỵ)炻㇨↮⼿䘬㭝㔠炻⯙㗗忁ᾳ㔠䘬⸛㕡㟡ᷳ㔜㔠䘬ỵ㔠ˤἳ⤪炻1156 ⎗ẍ↮ㆸ 11’56 ℑ㭝炻⬫䘬 ⸛㕡㟡㚱ℑỵ㔜㔠烊85264 ⎗ẍ↮ㆸ 8’52’64 ᶱ㭝炻⬫䘬⸛㕡㟡㚱ᶱỵ㔜㔠ˤ ℞㫉炻天䡢⭂忁ᾳ㬋㔜㔠䘬䬿埻⸛㕡㟡ᷳ㚨檀ỵᶲ䘬㔠ˤ ⛐ℑᾳ㬋㔠ᷕ炻庫⣏䘬㔠⸛㕡ḇ庫⣏炻⍵忶Ἦ炻庫⣏䘬㔠ᷳ 䬿埻⸛㕡㟡ḇ庫⣏ˤ㟡㒂忁ᾳ⿏岒炻⎗ẍ䡢⭂ᶨᾳ㔠䘬䬿埻⸛㕡㟡ᷳ㚨檀ỵᶲ䘬㔠ˤἳ⤪ 11’56 䘬ⶎ怲䫔ᶨ㭝㗗 11炻11 ⛐ 2 3 冯42 ᷳ攻炻䓙㬌⎗䞍炻1156 ⛐ 2 30 冯40 ᷳ攻炻㇨ẍ2 1156 䘬䬿埻⸛㕡㟡ᷳ㚨檀ỵᶲ䘬㔠㗗 3烊8’52’64 䘬ⶎ怲䫔ᶨ㭝㗗 8炻8 ⛐ 2 2 冯32 ᷳ攻炻䓙㬌⎗䞍炻85264 ⛐ 2 200 冯300 ᷳ攻炻㇨ẍ2 85264 䘬䬿埻 ⸛㕡㟡ᷳ㚨檀ỵᶲ䘬㔠㗗 2ˤ

(34)

䫔ᶱ炻䡢⭂㬋㔜㔠䘬⸛㕡㟡ᷳ℞⬫ỵᶲ䘬㔠炻ᶳ朊ㆹᾹ忂忶 ἳ柴Ἦ䞼䨞䡢⭂䘬㕡㱽ˤ ἳ⤪炻㯪 1156 ˤ ㆹᾹ⶚䴻䞍忻 1156 㚱ℑỵ㔜㔠炻㚨檀ỵᶲ䘬㔠㗗 3炻⌛⸛ 㕡㟡䘬⋩ỵ㔠㗗 3ˤ⤪㝄䓐 a ẋ堐忁ᾳ⸛㕡㟡䘬ᾳỵᶲ䘬㔠炻恋 湤⸛㕡㟡⯙⎗ẍ⮓ㆸ 30 a+ 䘬⼊⺷ˤ㕤㗗 2 2 2 1156 (30 ) 30 2 30 a a a = + = + × + ⽆ 1156 㷃⍣ 2 30 = 900炻⼿ ⯙㗗婒炻 2 256 = 2 30× a+ a = (2 30× + a a) ˤ㟡㒂忁ᾳ斄Ὢ炻ㆹᾹ⎗ ẍ㯪↢ᾳỵ㔠 aˤ 䓙㕤 a 㗗ᾳỵ㔠炻 2 30× 㭼 a ⣏⼿⣂炻㇨ẍ㈲ 2 30 a× + 䚳役Ụ㕤 2 30× 炻䓐 2 30× ⍣娎昌 256炻⼿⇘娎⓮ 4ˤ 天䡢⭂ a 䘬ῤ㗗⏎䫱㕤 4炻⎒天妰䬿䔞a = 㗪炻4 (2 30× + a a) 䘬ῤ㗗ᶵ㗗 256ˤ䓙 (60 4) 4 256+ = 炻㇨ẍ a 䘬ῤ〘㗗 4炻⚈㬌 1156 = 34ˤ ᶲ朊㇨婒䘬妰䬿忶䦳炻⎗ẍ⮓ㆸᶳ朊䘬寶⺷烉忁墉检㔠㗗暞炻堐䣢攳䘬䚉炻ḇ⯙㗗342 =1156炻1156 㗗ᶨ ᾳ⬴ℐ⸛㕡㔠ˤᶨ凔⛘炻⤪㝄ᶨᾳ㬋㔠〘⤥㗗⎎ᶨᾳ㚱䎮㔠䘬⸛ 㕡炻忁ᾳ㬋㔠⎓ ֹӄѳБኧˤ 1 1 5 6 炼 9 0 0 2 5 6 2 (30+ a) 2 30 2 2 30a× + a 11’56 3 4 9 2 56 2 56 2 30× = 60 4 64 0 為

(35)

䫮䬿攳⸛㕡㺼䬿䘬ᷣ天㬍樇㗗烉 1. ㈲塓攳㕡䘬㔜㔠炻⽆ᾳỵ崟⎹ⶎ㭷昼ℑỵ䁢ᶨ㭝炻䓐㐯嘇↮攳烊 2. 㟡㒂ⶎ怲䫔ᶨ㭝墉䘬㔠炻㯪⼿䬿埻⸛㕡㟡䘬㚨檀ỵᶲᷳ 㔠烊 3. ⽆䫔ᶨ㭝墉䘬㔠㷃⍣㯪⼿ᷳ㚨檀ỵᶲ㔠䘬⸛㕡炻⛐⬫Ᾱ 䘬ⶖᷳ⎛怲⮓ᶲ䫔Ḵ㭝㔠ἄ䁢䫔ᶨᾳ检㔠烊 4. ㈲㯪⼿䘬⇅⓮Ḁẍ 20 ⍣娎昌䫔ᶨᾳ检㔠炻㇨⼿䘬㚨⣏ 㔜㔠ἄ䁢娎⓮(⤪㝄忁ᾳ㚨⣏㔜㔠⣏㕤ㆾ䫱㕤 10炻⯙䓐 9 娎⓮)烊 5. 䓐⇅⓮䘬 20 ᾵≈ᶲ忁ᾳ娎⓮ℵḀẍ娎⓮炻⤪㝄㇨⼿䘬䧵⮷㕤ㆾ䫱㕤检㔠炻忁ᾳ娎⓮⯙㗗䬿埻⸛㕡㟡䘬䫔Ḵỵ 㔠烊⤪㝄㇨⼿䘬䧵⣏㕤检㔠炻⯙㈲娎⓮徸㫉䷖⮷ℵ娎炻䚜⇘䧵⮷㕤ㆾ䫱㕤检㔠䁢㬊烊 6. 䓐⎴㧋䘬㕡㱽炻两临㯪䬿埻⸛㕡㟡䘬℞⬫⎬ỵᶲᷳ㔠ˤ Ȝּ 1ȝ 㯪 (1) 1444 烊 (2) 85264 ˤ

ś

ྋ !!! (1) (2) ɺ 1444 =38 ɺ ₈₅₂₆₄ = ₂₉₂ ݙཀǺἳ 1 䫔(1)⮷柴ᷕ 544 昌 60 ⼿娎⓮ 9炻Ữ㗗 69 9× 䘬䧵⣏㕤 544炻㇨ẍ娎⓮㓡䓐 8烊ἳ 1 䫔(2)⮷柴ᷕ 452 昌 40 ⼿娎⓮ 11炻Ữ㗗娎⓮⎒傥㗗ᶨỵ㔠炻㇨ẍ㓡䓐 9ˤ 14’44 3 8 9 5 44 5 44 2 30× = 60 8 68 0 為 11 64 49 4 52 8’52’64 2 9 4 4 41 582 0 2 11 64

(36)

Ȝּ 2ȝ 㯪 10404 ˤ

ś

ྋ !! ɺ 10404 =102 ݙཀǺ⛐䫔ᶨ㬍妰䬿ᷕ炻1 㷃 1炻检 0ˤ⮓ᶳ塓攳㕡㔠䘬ᶳᶨ㭝⼿ 4炻䓐 20 昌 4 ᶵ⣈⓮ 1炻⛐䬿埻⸛㕡㟡䘬䫔Ḵỵ⮓ 0炻ℵ ㈲塓攳㕡㔠䘬䫔ᶱ㭝⮓ᶳἮ炻⼿ 404ˤ䓐 200 ⍣昌 404 ⼿娎⓮ 2ˤ 㬋⮷㔠䘬攳⸛㕡炻⎗ẍầ䄏ᶲ朊䘬㕡㱽Ἦ㺼䬿炻ᶵ⎴䘬㗗↮ 㭝ᷳ㗪῁炻天⽆⮷㔠溆⎹ⶎˣ⎹⎛㭷昼ℑỵ䓐㐯嘇↮攳炻㚨⼴ᶨ 㭝⤪㝄⎒㚱ᶨỵ炻⽭枰㶣暞墄ㆸℑỵˤἳ⤪ 0.3249 ↮㭝⼴ㅱ娚㗗 0.32’49炻232.5625 ↮㭝⼴ㅱ娚㗗 2’32.56’25炻321.512 ↮㭝⼴ㅱ娚㗗 3’21.51’20ˤ怬天㲐シ㇨⼿䳸㝄䘬⮷㔠溆⽭枰冯塓攳㕡㔠䘬溆 ⮵滲ˤ Ȝּ 3ȝ 㯪 (1) 0.3249 烊 (2) 232.5625 ˤ

ś

ྋ !!! (1) (2) ɺ 0.3249 = 0.57 ɺ _232.5625 =_15.25 4 04 1’04’04 1 0 1 4 04 202 0 2 7 49 0.32’49 0. 5 25 7 49 107 0 7 7 56 25 1 32 2’32.56’25 1 5. 1 1 25 302 0 2 6 04 5 1 52 25 1 52 25 3045

(37)

⇵朊ᶱᾳἳ⫸ᷕ㇨䴎↢䘬塓攳㕡㔠悥㗗⬴ℐ⸛㕡㔠ˤ栆Ụ⛘ ḇ⎗ẍ㯪↢朆⬴ℐ⸛㕡㔠䘬役Ụ⸛㕡㟡ˤ Ȝּ 4ȝ 㯪 (1) 12.5 䘬役Ụῤ(䱦䡢⇘ 0.01)烊 (2) 2 䘬役Ụῤ(䱦䡢⇘ 0.0001)ˤ

ś

ྋ !!! (1) (2) ɺ 12.5 ≈ 3.54 ɺ ₂ ≈_1.4142 ἳ 4 ᷕ䘬 12.5 冯 2 悥㗗朆⬴ℐ⸛㕡㔠炻䫮䬿攳㕡㗪⎗ẍὅ䄏天㯪䘬䱦䡢⹎怑䔞墄暞炻ℵ忚埴攳㕡忳䬿ˤ⽭枰㲐シ墄暞㗪ㅱ娚 ἧ⮷㔠悐↮傥⣈↮ㆸ㭷ℑỵᶨ㭝ˤ 3 91 00 25 00 65 3 50 12.50’00’00 5 3. 9 3 25 703 37 75 3 21 09 5 3 53 25 7065 1 19 00 4 00 24 1 00 2.00’00’00’00’00 1. 4 1 96 281 10 07 59 1 2 81 4 1 12 96 2824 2 1 6 04 00 5 65 64 28282 38 36 00 28 28 41 282841

第九章 數的開方

注意：