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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

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
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
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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