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Eksponente - graad 11

Inleiding

In Graad 11 het ons eksponensiële getalle bestudeer en ons het die ses wette geleer wat bewerking met eksponensiële getalle baie makliker gemaak het. Daar is een wet wat ons nie in Graad 11 gedoen het nie. Dit sal ons hier beskryf.

Wette van eksponente

In Graad 11 het ons net met indekse gewerk wat in heelgetalle was. Wat gebeur is die indeks nie 'n heelgetal is nie, maar 'n rasionele getal? Dit lei ons na die finale wet van eksponente,

a m n = a m n

Eksponensiële wet 7: a m n = a m n

Ons sê dat x is 'n n de wortel van b as x n = b en ons kan skryf x = b n . n de wortels geskryf met die radikale simbool, , word verwys as wortelvorme. Byvoorbeeld, ( - 1 ) 4 = 1 , so - 1 is 'n 4de wortel van 1. Waneer ons wet 6 gebruik sien ons dat,

( a m n ) n = a m n × n = a m

dus a m n moet 'n n de wortel van a m wees. Ons kan dus sê,

a m n = a m n

Byvoorbeeld,

2 2 3 = 2 2 3

'n Getal mag nie altyd 'n rasionele n de wortel hê nie. Byvoorbeeld, as n = 2 en a = - 1 , dan is daar geen rasionele getal so dat x 2 = - 1 omdat x 2 0 vir alle rasionele getalle van x .

Komplekse getalle

Daar is getalle wat probleme kan oplos soos x 2 = - 1 , maar dit is buite die omvang van hierdie boek. Hulle word genoem komplekse getalle .

Dit is ook moentlik vir meer as een n de wortel vir 'n gegewe getal om te bestaan. Byvoorbeeld, ( - 2 ) 2 = 4 en 2 2 = 4 , so beide -2 en 2 is 2de (vierkants) wortels van 4. Gewoonlik, as daar meer as een wortel is, dan kies ons die positiewe reële getal en ons gaan aan.

Vereenvoudig sonder die gebruik van 'n sakrekenaar:

5 4 - 1 - 9 - 1 1 2
  1. = 5 1 4 - 1 9 1 2
  2. = 5 1 ÷ 9 - 4 36 1 2 = 5 1 × 36 5 1 2 = ( 6 2 ) 1 2
  3. = 6

Vereenvoudig:

( 16 x 4 ) 3 4
  1. = ( 2 4 x 4 ) 3 4
  2. = 2 4 × 3 4 . x 4 × 3 4 = 2 3 . x 3 = 8 x 3

Toepassing van wette

Gebruik al die wette om:

  1. Vereenvoudig:
    (a) ( x 0 ) + 5 x 0 - ( 0 , 25 ) - 0 , 5 + 8 2 3 (b) s 1 2 ÷ s 1 3
    (c) 12 m 7 9 8 m - 11 9 (d) ( 64 m 6 ) 2 3
  2. Her-skryf die volgende uitdrukking as 'n krag van x :
    x x x x x

Eksponensiële in die werklike wêreld

In Graad 10 Finansies, het julle eksponensiële gebruik om verskillende tipe rente te bereken. Byvoorbeeld op 'n spaarrekening of op 'n lening en saamgestelde groei.

'n Spesifike tipe bakterieë het 'n baie hoë eksponensiële groei koers teen 80% elke uur. As daar 10 bakterieë is, bepaal hoeveel daar sal wees na 5 uur, 'n dag en na 1 week?

  1. Daarom, in hierdie geval:

    P o p u l a t i o n = 10 ( 1 , 8 ) n , waar n = aantel ure

  2. P o p u l a t i o n = 10 ( 1 , 8 ) 5 = 189

  3. P o p u l a t i o n = 10 ( 1 , 8 ) 24 = 13 382 588

  4. P o p u l a t i o n = 10 ( 1 , 8 ) 168 = 7 , 687 × 10 43

    Let op dat hierdie antwoord in wetenskaplike notasie aangedei woord want dit is 'n baie groot getal.

'n Spesifike soort van uiters skaars diep water vis het 'n baie lang leeftyd en het slede kinders. As daar 'n totaal van 821 van hierdie tipe vis is en hulle groei koers is 2% per mannd, hoeveel sal daar wees by die helfte van 'n gegewe jaar? Wat sal de bevolking wees in 10 jaar en in 'n 100 jaar wees?

  1. Daarom, in hierdie geval:

    P o p u l a t i o n = 821 ( 1 , 02 ) n , waar n = aantal maande

  2. P o p u l a t i o n = 821 ( 1 , 02 ) 6 = 925

  3. P o p u l a t i o n = 821 ( 1 , 02 ) 120 = 8 838

  4. P o p u l a t i o n = 821 ( 1 , 02 ) 1 200 = 1 , 716 × 10 13

    Let op dat hierdie antwoord in wetenskaplike notasie aangedui woord want dit is 'n baie groot getal.

Einde van hoofstuk oefeninge

  1. Vereenvoudig so ver as moontlik:
    1. 8 - 2 3
    2. 16 + 8 - 2 3
  2. Vereenvoudig:
    (a) ( x 3 ) 4 3 (b) ( s 2 ) 1 2
    (c) ( m 5 ) 5 3 (d) ( - m 2 ) 4 3
    (e) - ( m 2 ) 4 3 (f) ( 3 y 4 3 ) 4
  3. Vereenvoudig so veel as wat jy kan:
    3 a - 2 b 15 c - 5 a - 4 b 3 c - 5 2
  4. Vereenvoudig so veel as wat jy kan:
    9 a 6 b 4 1 2
  5. Vereenvoudig so veel as wat jy kan:
    a 3 2 b 3 4 16
  6. Vereenvoudig:
    x 3 x
  7. Vereenvoudig:
    x 4 b 5 3
  8. Herskryf die volgende uitdrukking as 'n krag van x :
    x x x x x x 3

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
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Berger describes sociologists as concerned with
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Source:  OpenStax, Siyavula textbooks: wiskunde (graad 11). OpenStax CNX. Sep 20, 2011 Download for free at http://cnx.org/content/col11339/1.4
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