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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

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Wiskunde graad 9. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col11055/1.1
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