<< Chapter < Page Chapter >> Page >

Wiskunde

Graad 9

Getalle

Module 2

Maak wiskunde makliker met eksponente

KLASWERK

  • Onthou jy nog hoe eksponente werk? Skryf neer wat “drie tot die mag sewe” beteken. Wat is die grondtal? Wat is die eksponent? Kan jy mooi verduidelik wat ’n mag is?
  • Hierdie deel het baie voorbeelde met getalle; gebruik jou sakrekenaar om hulle uit te werk sodat jy vertroue in die metodes kan ontwikkel.

1. DEFINISIE

2 3 = 2 × 2 × 2 en a 4 = a × a × a × a en b × b × b = b 3

ook

( a+ b ) 3 = ( a + b ) × ( a + b ) × ( a + b ) en 2 3 4 = 2 3 × 2 3 × 2 3 × 2 3 size 12{ left ( { {2} over {3} } right ) rSup { size 8{4} } = left ( { {2} over {3} } right ) times left ( { {2} over {3} } right ) times left ( { {2} over {3} } right ) times left ( { {2} over {3} } right )} {}

1.1 Skryf die volgende uitdrukkings in uitgebreide vorm:

4 3 ( p +2) 5 a 1 (0,5) 7 b 2 × b 3

1.2 Skryf hierdie uitdrukkings as magte:

7 × 7 × 7 × 7 y × y × y × y × y –2 × –2 × –2 ( x + y ) × ( x + y ) × (x + y ) × ( x + y )

1.3 Antwoord sonder om dit uit te werk: Is (–7) 6 dieselfde as –7 6 ?

  • Gebruik nou ’n sakrekenaar en kyk of die twee waardes dieselfde is.
  • Vergelyk ook die volgende pare deur eers te raai wat die antwoord gaan wees, en dan met jou sakrekenaar te kyk hoe goed jy geskat het.

–5 2 en (–5) 2 –12 5 en (–12) 5 –1 3 en (–1) 3

  • Jy behoort nou ’n goeie idee te hê hoe hakies antwoorde beïnvloed – skryf dit neer sodat jy dit sal onthou en in die toekoms kan gebruik wanneer die probleme moeiliker word.
  • Ons som hierdie deel op in ’n definisie:

a r = a × a × a × a × . . . (daar moet r a ’s wees, en r moet ’n natuurlike getal wees)

  • Van nou af moet jy die belangrikste magte begin memoriseer:

2 2 = 4; 2 3 = 8; 2 4 = 16; ens. 3 2 = 9; 3 3 = 27; 3 4 = 81; ens. 4 2 = 16; 4 3 = 64; ens.

Die meeste eksponentsomme moet sonder ’n sakrekenaar gedoen word.

2 VERMENIGVULDIGING

  • Onthou jy nog dat g 3 × g 8 = g 11 ? Kernwoorde: vermenigvuldig ; dieselfde grondtal

2.1 Vereenvoudig: (moenie uitgebreide vorm gebruik nie).

7 7 × 7 7 (–2) 4 × (–2) 13 ( ½ ) 1 × ( ½ ) 2 × ( ½ ) 3 ( a+b ) a × ( a+b ) b

  • Ons vermenigvuldig magte met enerse grondtalle volgens hierdie reël:

a x × a y = a x+y ook = a x a y = a y a x a x + y size 12{ size 11{a rSup { size 8{ size 7{x+y}} } } size 12{ {}=}a rSup { size 8{x} } size 12{ times }a rSup { size 8{y} } size 12{ {}=}a rSup { size 8{y} } size 12{ times }a rSup { size 8{x} } } {} , bv. 8 14 = 8 4 × 8 10 size 12{8 rSup { size 8{"14"} } =8 rSup { size 8{4} } times 8 rSup { size 8{"10"} } } {}

3. DELING

  • 4 6 4 2 = 4 6 2 = 4 4 size 12{ { {4 rSup { size 8{6} } } over {4 rSup { size 8{2} } } } =4 rSup { size 8{6 - 2} } =4 rSup { size 8{4} } } {} is hoe dit werk. Kernwoorde: deel ; dieselfde grondtal

3.1 Probeer hierdie: a 6 a y size 12{ { { size 11{a rSup { size 8{6} } }} over { size 12{a rSup { size 8{y} } } } } } {} 3 23 3 21 size 12{ { {3 rSup { size 8{"23"} } } over {3 rSup { size 8{"21"} } } } } {} a + b p a + b 12 size 12{ { { left ( size 11{a+b} right ) rSup { size 8{p} } } over { size 12{ left (a+b right ) rSup { size 8{"12"} } } } } } {} a 7 a 7 size 12{ { { size 11{a rSup { size 8{7} } }} over { size 12{a rSup { size 8{7} } } } } } {}

  • Die reël wat ons gebruik vir deling van magte is: a x a y = a x y size 12{ { { size 11{a rSup { size 8{x} } }} over { size 12{a rSup { size 8{y} } } } } size 12{ {}=}a rSup { size 8{x - y} } } {} .

Ook a x y = a x a y size 12{ size 11{a rSup { size 8{x - y} } } size 12{ {}= { {a rSup { size 8{x} } } over { size 12{a rSup { size 8{y} } } } } }} {} , bv. a 7 = a 20 a 13 size 12{ size 11{a rSup { size 8{7} } } size 12{ {}= { {a rSup { size 8{"20"} } } over { size 12{a rSup { size 8{"13"} } } } } }} {}

4. VERHEFFING VAN ’n MAG TOT ’n MAG

  • bv. 3 2 4 size 12{ left (3 rSup { size 8{2} } right ) rSup { size 8{4} } } {} = 3 2 × 4 size 12{3 rSup { size 8{2 times 4} } } {} = 3 8 size 12{3 rSup { size 8{8} } } {} .

4.1 Doen die volgende:

  • Die reël werk so: a x y = a xy size 12{ left (a rSup { size 8{x} } right ) rSup { size 8{y} } =a rSup { size 8{ ital "xy"} } } {} ook a xy = a x y = a y x size 12{ size 11{a rSup { size 8{ bold "xy"} } } size 12{ {}= left (a rSup { size 8{x} } right ) rSup { size 8{y} } } size 12{ {}= left (a rSup { size 8{y} } right ) rSup { size 8{x} } }} {} , bv. 6 18 = 6 6 3 size 12{6 rSup { size 8{"18"} } = left (6 rSup { size 8{6} } right ) rSup { size 8{3} } } {}

5. DIE MAG VAN ’n PRODUK

  • So werk dit:

(2 a ) 3 = (2 a ) × (2 a ) × (2 a ) = 2 × a × 2 × a × 2 × a = 2 × 2 × 2 × a × a × a = 8 a 3

  • Dit word gewoonlik in twee stappe gedoen, nl.: (2 a ) 3 = 2 3 × a 3 = 8 a 3

5.1 Doen self hierdie: (4 x ) 2 ( ab ) 6 (3 × 2) 4 ( ½ x ) 2 ( a 2 b 3 ) 2

  • Dis duidelik dat die eksponent aan elke faktor in die hakies behoort.
  • Hier is die reël: ( ab ) x = a x b x ook a p b p = ab b size 12{ size 11{a rSup { size 8{p} } } size 12{ times }b rSup { size 8{p} } size 12{ {}= left ( bold "ab" right ) rSup { size 8{b} } }} {} bv. 14 3 = 2 × 7 3 = 2 3 7 3 size 12{"14" rSup { size 8{3} } = left (2 times 7 right ) rSup { size 8{3} } =2 rSup { size 8{3} } 7 rSup { size 8{3} } } {} en 3 2 × 4 2 = 3 × 4 2 = 12 2 size 12{3 rSup { size 8{2} } times 4 rSup { size 8{2} } = left (3 times 4 right ) rSup { size 8{2} } ="12" rSup { size 8{2} } } {}

6. DIE MAG VAN ’n BREUK

  • Dis baie dieselfde as die mag van ’n produk. a b 3 = a 3 b 3 size 12{ left ( { { size 11{a}} over { size 11{b}} } right ) rSup { size 8{3} } size 12{ {}= { {a rSup { size 8{3} } } over { size 12{b rSup { size 8{3} } } } } }} {}

6.1 Doen hierdie, maar wees versigtig: 2 3 p size 12{ left ( { {2} over {3} } right ) rSup { size 8{p} } } {} 2 2 3 size 12{ left ( { { left ( - 2 right )} over {2} } right ) rSup { size 8{3} } } {} x 2 y 3 2 size 12{ left ( { { size 11{x rSup { size 8{2} } }} over { size 12{y rSup { size 8{3} } } } } right ) rSup { size 8{2} } } {} a x b y 2 size 12{ left ( { { size 11{a rSup { size 8{ - x} } }} over { size 12{b rSup { size 8{ - y} } } } } right ) rSup { size 8{ - 2} } } {}

  • Weer behoort die eksponent aan beide die teller en die noemer.
  • Die reël: a b m = a m b m size 12{ left ( { { size 11{a}} over { size 11{b}} } right ) rSup { size 8{m} } size 12{ {}= { {a rSup { size 8{m} } } over { size 12{b rSup { size 8{m} } } } } }} {} en a m b m = a b m size 12{ { { size 11{a rSup { size 8{m} } }} over { size 12{b rSup { size 8{m} } } } } size 12{ {}= left ( { {a} over { size 12{b} } } right ) rSup { size 8{m} } }} {} bv. 2 3 3 = 2 3 3 3 = 8 27 size 12{ left ( { {2} over {3} } right ) rSup { size 8{3} } = { {2 rSup { size 8{3} } } over {3 rSup { size 8{3} } } } = { {8} over {"27"} } } {} en a 2x b x = a 2 x b x = a 2 b x size 12{ { { size 11{a rSup { size 8{2x} } }} over { size 12{b rSup { size 8{x} } } } } = { { left ( size 11{a rSup { size 8{2} } } right ) rSup { size 8{x} } } over { size 12{b rSup { size 8{x} } } } } size 12{ {}= left ( { {a rSup { size 8{2} } } over { size 12{b} } } right ) rSup { size 8{x} } }} {}

einde van KLASWERK

TUTORIAAL

  • Pas hierdie reëls saam toe om die volgende uitdrukkings te vereenvoudig — sonder ’n sakrekenaar.

1. a 5 a 7 a a 8 size 12{ { { size 11{a rSup { size 8{5} } } size 12{ times }a rSup { size 8{7} } } over { size 12{a size 12{ times }a rSup { size 8{8} } } } } } {}

2. x 3 y 4 x 2 y 5 x 4 y 8 size 12{ { { size 11{x rSup { size 8{3} } } size 12{ times }y rSup { size 8{4} } size 12{ times }x rSup { size 8{2} } y rSup { size 8{5} } } over { size 12{x rSup { size 8{4} } y rSup { size 8{8} } } } } } {}

3. a 2 b 3 c 2 ac 2 2 bc 2 size 12{ left ( size 11{a rSup { size 8{2} } b rSup { size 8{3} } c} right ) rSup { size 8{2} } size 12{ times left ( bold "ac" rSup { size 8{2} } right ) rSup { size 8{2} } } size 12{ times left ( bold "bc" right ) rSup { size 8{2} } }} {}

4. a 3 b 2 a 3 a b 5 b 4 ab 3 size 12{ size 11{a rSup { size 8{3} } } size 12{ times }b rSup { size 8{2} } size 12{ times { {a rSup { size 8{3} } } over { size 12{a} } } } size 12{ times { {b rSup { size 8{5} } } over { size 12{b rSup { size 8{4} } } } } } size 12{ times left ( bold "ab" right ) rSup { size 8{3} } }} {}

5. 2 xy × 2 x 2 y 4 2 x 2 y 3 2 xy 3 size 12{ left (2 size 11{ bold "xy"} right ) times left (2 size 11{x rSup { size 8{2} } y rSup { size 8{4} } } right ) rSup { size 8{2} } size 12{ times left ( { { left (x rSup { size 8{2} } y right ) rSup { size 8{3} } } over { size 12{ left (2 bold "xy" right ) rSup { size 8{3} } } } } right )}} {}

6. 2 3 × 2 2 × 2 7 8 × 4 × 8 × 2 × 8 size 12{ { {2 rSup { size 8{3} } times 2 rSup { size 8{2} } times 2 rSup { size 8{7} } } over {8 times 4 times 8 times 2 times 8} } } {}

einde van TUTORIAAL

Nog ’n paar reëls

KLASWERK

1 Beskou hierdie geval: = a 5 3 = a 2 a 5 a 3 size 12{ { { size 11{a rSup { size 8{5} } }} over { size 12{a rSup { size 8{3} } } } } size 12{ {}=}a rSup { size 8{5 - 3} } size 12{ {}=}a rSup { size 8{2} } } {}

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Wiskunde graad 9. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col11055/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Wiskunde graad 9' conversation and receive update notifications?

Ask