Laws of Integral Indices

Transcription

1 A Lws of Itegrl Idices A. Positive Itegrl Idices I, is clled the se, is clled the idex lso clled the expoet. mes times.... Exmple Simplify 5 6 c Solutio c 6 Exmple Simplify Solutio

2 The results i Exmple d Exmple illustrte the followig lws of idices: If m d re positive itegers, the i m m + ii m m, 0 d m > iii m m iv v, 0 Exmple Simplify c d 9 Solutio c d

3 Checkpoit Simplify c 5 p q r p q r p p p d p p q q q p

4 A. Zero d Negtive Itegrl Idices We re ow goig to exted the ide of expoets to zero d egtive itegrl idices i such wy tht the lws of idices hold for ll itegrl idices. Cosider the followig ptters: The idices of 0 re decresig ech step y ?? 0?? 0?? These umers re decresig ech step to 0 of the previous umer. The idices of re decresig ech step y. These umers re decresig ech step to of the previous umer. From the ptters, we oserve tht By the lw m m+, we hve if if if 0 Thus, if 0 d is iteger, we defie 0

5 Now we hve the followig lws of idices: If m d re positive itegers, the i ii iii iv m m+ m m, 0 m m d m > v, 0 Exmple Fid the vlues of Solutio Exmple 5 Simplify 0 c m c Give the swers with positive idices. d m Solutio 0 c m m m m m m c d m 5

6 Checkpoit Fid the vlues of c d 5 Checkpoit Simplify the followig: Give the swers with positive idices. k k c p s 6

7 Checkpoit Simplify the followig: k k + c + Give the swers with positive idices. 7

8 A. Nottios for Vrious Numerl Systems A. Numerl Systems roud Us I everydy life, we prefer usig mesurig uits i the metric system 十進制 to simplify clcultio d coversio. The metric system is deciml or dery system, the coversio fctors re either 0 or powers of 0. For exmple, uits for mesurig legths re metre m, cetimetre cm, millimetre mm, etc., where m 00 cm d cm 0 mm. As well s metric or deciml system, other umerl systems re used i everydy life. The followig tle shows some everydy exmples of o-metric systems. No-metric system Exmple Remrk Ctties 斤 d tels 兩 or pouds 磅 d ouces 安士 re used i mesurig weights. Hexdeciml Bse 6 ctty 6 tels poud 6 ouces Feet 尺 d iches 寸 re used i mesurig heights. Duodeciml Bse foot iches Biry Bse The umerl system is used i iterl opertios of computers. Computers store iformtio i the form of comitios of 0 d. 8

9 B. Plce Vlue Plce Vlue of Deciml Numers I the deciml system of umertio, ech digit 數字 c tke y oe of the followig Aric umerls: 0,,,,, 5, 6, 7, 8 d 9. For exmple, we c write two hudred d thirtee s. I deciml umer, the positio tht ech umerl tkes hs certi vlue, which is clled the plce vlue 位值. For exmple, i the umer, the plce vlue of ech digit from the right to left is s follows: The plce vlue of the first digit,, is oe ; the plce vlue of the secod digit,, is te 0; the plce vlue of the third digit,, is hudred 00. Moreover, plce vlues icrese i powers of 0 from right to left. The plce vlue of ech digit is 0 times tht of the umer to its right. Hudreds Tes Oes Digit Plce vlue We c use umericl expressio which shows the plce vlue of ech digit to represet y umer. e.g Plce vlue of Plce vlue of Plce vlue of We cll the umericl expressio tht shows the plce vlue of ech digit the expded form 展開式. By usig the expded form, we c show clerly the reltio etwee the plce vlues of differet digits. For the deciml umer 0, its expded form is ; 0 is clled the plce holder 補位數字. Although 0 mes othig i qutity, it plys very importt role here. It is ecuse we c determie the plce vlues of other digits oly fter 0 hs tke its positio. 9

10 Exmple 6 Cosider the umer 00. Wht re the plce vlues of d? Write 00 i the expded form. Solutio Plce vlue of 0 Plce vlue of Checkpoit 5 Write the followig deciml umers i the expded form c 85 0 Checkpoit 6 Represet ech of the followig expressios s deciml umer c

11 Plce vlue of Biry Numers d Hexdeciml Numers Other th the deciml system, the iry d the hexdeciml systems re two commo umerl systems. The iry system hs two umerls: 0 d. The hexdeciml system hs sixtee umerls: 0,,,,, 5, 6, 7, 8, 9, A, B, C, D, E d F. The tle shows the deciml umerls correspodig to the umerls A to F i the hexdeciml system. Hexdeciml Deciml A 0 B C D E F 5 I order to distiguish umers i differet systems, we idicte the se of the umer t its ottom right corer. e.g. 6 0 is deciml umer. 00 is iry umer. A 6 is hexdeciml umer. Like the deciml umers, plce vlues of iry umers d hexdeciml umers icrese from right to left. For exmple: Plce vlues of iry umer 00 : Biry 0 0 Plce vlue The expded form of Plce vlues of iry umer A 6 : Hexdeciml A Plce vlue 6 The expded form of A Note: All iry umers d hexdeciml umers expressed i the expded form i.e. the right-hd side of the equl sig re writte s deciml umers.

12 Checkpoit 7 Write dow the plce vlue of the digit 0 i ech of the followig umers c 5E06 6 Checkpoit 8 Write the followig iry umers i the expded form. 0 c Checkpoit 9 Write the followig hexdeciml umers i the expded form A 6 c F69 6

13 Exmple 7 Represet the expressio + s iry umer. Represet the expressio s hexdeciml umer. Solutio F0 6 Checkpoit 0 Represet ech of the followig expressios s iry umer c Checkpoit Represet ech of the followig expressios s hexdeciml umer c

14 A. Coversios Betwee Numers of Differet Numerl Systems A. Covertig Biry Numers or Hexdeciml Numers ito Deciml Numers We c covert the origil iry or hexdeciml umer ito deciml umer y evlutig the expded form of tht umer. Exmple 8 Covert 0 ito deciml umer. Solutio Exmple 9 Covert D 6 ito deciml umer. Solutio D Checkpoit Covert ech of the followig iry umers ito deciml umer. 0 00

15 Checkpoit Covert ech of the followig hexdeciml umers ito deciml umer. A 508 B. Covertig Biry Numers or Hexdeciml Numers ito Deciml Numers We c form iry umer if we kow its expded form. For exmple, Whe we covert deciml umer, such s, ito iry umer, how do we fid the expded form of iry umer from the deciml umer? The method is show s follows: Divide the deciml umer cotiuously y util the quotiet is, i.e. less th. Red the fil quotiet i.e. first, the ll the remiders from ottom to top successively. For exmple, to covert 0 ito its expded form of iry umer, Remider 6 0 Quotiet less th 6 0 Red the iry umer Thus, the expded form of 0 of iry umer is , d

16 Exmple 0 Covert 0 ito iry umer. Solutio Similrly, we c lso use the cotiul divisio to covert deciml umer ito hexdeciml umer. The method is to divide the deciml umer cotiully y 6 util the quotiet is less th 6. Exmple Covert 00 0 ito hexdeciml umer. Solutio D 6 6

17 Checkpoit I ech of the followig, i covert the deciml umer ito iry umer; ii covert the deciml umer ito hexdeciml umer c 6 0 7

18 Exercise A Lws of Itegrl Idices A.. Simplify the followig: 6 k k k 8 p p p c c c d c c e c f g q r h x y z i 5 h 9h j h k c 6 c l 6 p p 6 p A.. Fid the vlues of the followig d give the swers i frctios. 8 8 c d 8 5 e 6 f g + + h. I ech of the followig, simplify the expressio d express the swer with positive idices. All letters give represet o-zero umers. c e m d c c f m m 0 0 g h c i k 5 j rs l r s s 8

19 A.. Represet ech of the followig expressios s deciml umer c d Represet ech of the followig expressios s iry umer c d Represet ech of the followig expressios s hexdeciml umer c d Write the followig umers of differet systems i the expded form c 0 d 0 e 00 f 000 g 5 6 h 0 6 i AD7 6 j BEA 6 A. 8. Covert the followig iry umers ito deciml umers. 000 c 0000 d Covert the followig hexdeciml umers ito deciml umers. AB 6 0E 6 c 69 6 d DC8 6 9

20 0. Covert the followig deciml umers ito iry umers c 6 0 d 5 0. Covert the followig deciml umers ito hexdeciml umers c d 0. Mr Yug ought sfe to keep some precious property t home. The pssword of the sfe ws 5-digit umer i the deciml system. He told his dughter, Dey, the pssword d sked her to fid wy to rememer it. Dey thought for while d used her me s hit to the pssword. She cosidered the first four lphets of hers me s -digit hexdeciml umer. The pssword of the sfe would e the umer otied y covertig this hexdeciml umer ito deciml umer. Fid the pssword. 0