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Inleiding

In hierdie hoofstuk sal jy leer van 'n eenvoudiger manier om uitdrukkings soos 2 × 2 × 2 × 2 te skryf. Dit staan bekend as eksponensiaalnotasie .

Definisie

Eksponensiaalnotasie is 'n kort manier om te skryf dat 'n getal meermale met homself vermenigvuldig word. Byvoorbeeld, eerder as om te skryf 5 × 5 × 5 , gebruik ons 5 3 om aan te dui dat die getal 5 drie maal met homself vermenigvuldig word en 'n mens sê "5 tot die mag 3". Soortgelyk is 5 2 dieselfde as 5 × 5 en 3 5 is 3 × 3 × 3 × 3 × 3 . Laat ons beter definieer hoe om eksponensiaalnotasie te gebruik.

Eksponensiaalnotasie

Eksponensiaalnotasie verwys na 'n getal wat geskryf word as

a n

waar n 'n heelgetal is en a enige reële getal is. Ons noem a die grondtal en n die eksponent .

a tot die mag n is

a n = a × a × × a ( n -keer )

Dit wil sê, a word n keer met homself vermenigvuldig.

Ons kan ook 'n negatiewe eksponent, - n , gebruik. In hierdie geval

a - n = 1 a × a × × a ( n -keer )
Eksponente

Indien n 'n ewe getal is, sal a n altyd 'n positiewe getal wees vir enige reële getal a , behalwe 0 . Byvoorbeeld, hoewel - 2 negatief is, is beide ( - 2 ) 2 = - 2 × - 2 = 4 en ( - 2 ) - 2 = 1 - 2 × - 2 = 1 4 positief.

Khan academy video oor eksponente 1 (in engels)

Khan academy video oor eksponente 2 (in engels)

Eksponentwette

Daar is heelwat eksponentwette wat ons kan gebruik om getalle met eksponente te vereenvoudig. Sommige van hierdie wette het ons reeds in vorige grade teëgekom, maar ons sal die volledige lys hier sien en elke wet verduidelik, sodat jy hulle kan verstaan en nie bloot memoriseer nie.

a 0 = 1 a m × a n = a m + n a - n = 1 a n a m ÷ a n = a m - n ( a b ) n = a n b n ( a m ) n = a m n

Eksponente, wet 1: a 0 = 1

Volgens die definisie van eksponensiaalnotasie is

a 0 = 1 , ( a 0 )

Byvoorbeeld, x 0 = 1 en ( 1 000 000 ) 0 = 1

Toepassing van wet 1: a 0 = 1 , ( a 0 )

  1. 16 0
  2. 16 a 0
  3. ( 16 + a ) 0
  4. ( - 16 ) 0
  5. - 16 0

Eksponente, wet 2: a m × a n = a m + n

Khan academy video oor eksponente 3 (in engels)

Die definisie van eksponensiaalnotasie wys dat

a m × a n = 1 × a × ... × a ( m -keer ) × 1 × a × ... × a ( n -keer ) = 1 × a × ... × a ( m + n -keer ) = a m + n

Byvoorbeeld,

2 7 × 2 3 = ( 2 × 2 × 2 × 2 × 2 × 2 × 2 ) × ( 2 × 2 × 2 ) = 2 7 + 3 = 2 10

Interessante feit

Hierdie eenvoudige wet is die rede waarom eksponente oorspronklik geskep is. Voor die dae van rekenaars moes vermenigvuldiging met potlood en papier gedoen word. Dit vat baie lank om vermenigvuldiging te doen, maar dit is vinnig en eenvoudig om getalle bymekaar te tel. Hierdie eksponentwet wys dat dit moontlik is om twee getalle te vermenigvuldig deur hulle eksponente bymekaar te tel (indien hulle dieselfde grondtal het). Hierdie ontdekking het wiskundiges baie tyd gespaar, wat hulle toe kon gebruik om iets meer produktiefs te doen.

Toepassing van wet 2: a m × a n = a m + n

  1. x 2 · x 5
  2. 2 3 . 2 4 [Neem kennis dat die grondtal (2) dieselfde bly.]
  3. 3 × 3 2 a × 3 2

Eksponente, wet 3: a - n = 1 a n , a 0

Die definisie van eksponensiaalnotasie vir 'n negatiewe eksponent wys dat

a - n = 1 ÷ a ÷ ... ÷ a ( n -keer ) = 1 1 × a × × a ( n -keer ) = 1 a n

Dit beteken dat 'n minus teken in die eksponent 'n alternatiewe manier is om aan te dui dat die hele eksponensiaal gedeel eerder asvermenigvuldig moet word.

Byvoorbeeld,

2 - 7 = 1 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1 2 7

Toepassing van wet 3: a - n = 1 a n , a 0

  1. 2 - 2 = 1 2 2
  2. 2 - 2 3 2
  3. ( 2 3 ) - 3
  4. m n - 4
  5. a - 3 · x 4 a 5 · x - 2

Eksponente, wet 4: a m ÷ a n = a m - n

Met Wet 3 het ons reeds besef dat 'n minusteken 'n manier is om te wys dat die eksponensiaal gedeel eerder as vermenigvuldig moetword. Wet 4 is basies 'n meer algemene manier om dieselfde stelling te maak. Ons verkry hierdie wet deur Wet 3 aan beide kante met a m te vermenigvuldig en dan Wet 2 te gebruik.

a m a n = a m a - n = a m - n

Byvoorbeeld,

2 7 ÷ 2 3 = 2 × 2 × 2 × 2 × 2 × 2 × 2 2 × 2 × 2 = 2 × 2 × 2 × 2 = 2 4 = 2 7 - 3

Khan academy video oor eksponente 4 (in engels)

Toepassing van wet 4: a m ÷ a n = a m - n

  1. a 6 a 2 = a 6 - 2
  2. 3 2 3 6
  3. 32 a 2 4 a 8
  4. a 3 x a 4

Eksponente, wet 5: ( a b ) n = a n b n

Die volgorde waarin twee getalle vermenigvuldig word, is onbelangrik. Dus,

( a b ) n = a × b × a × b × ... × a × b ( n -keer ) = a × a × ... × a ( n -keer ) × b × b × ... × b ( n -keer ) = a n b n

Byvoorbeeld,

( 2 · 3 ) 4 = ( 2 · 3 ) × ( 2 · 3 ) × ( 2 · 3 ) × ( 2 · 3 ) = ( 2 × 2 × 2 × 2 ) × ( 3 × 3 × 3 × 3 ) = ( 2 4 ) × ( 3 4 ) = 2 4 3 4

Toepassing van wet 5: ( a b ) n = a n b n

  1. ( 2 x y ) 3 = 2 3 x 3 y 3
  2. ( 7 a b ) 2
  3. ( 5 a ) 3

Eksponente, wet 6: ( a m ) n = a m n

Dit is moontlik om die eksponensiaal van 'n eksponensiaal te bereken. Die eksponensiaal van 'n getal is 'n reële getal. So, selfs al klink die eerste sin ingewikkeld, beteken dit bloot dat 'n mens die eksponensiaal van 'n getal bereken en dan die eksponensiaal van die resultaat bereken.

( a m ) n = a m × a m × ... × a m ( n -keer ) = a × a × ... × a ( m × n -keer ) = a m n

Byvoorbeeld,

( 2 2 ) 3 = ( 2 2 ) × ( 2 2 ) × ( 2 2 ) = ( 2 × 2 ) × ( 2 × 2 ) × ( 2 × 2 ) = ( 2 6 ) = 2 ( 2 × 3 )

Toepassing van wet 6: ( a m ) n = a m n

  1. ( x 3 ) 4
  2. [ ( a 4 ) 3 ] 2
  3. ( 3 n + 3 ) 2

Vereenvoudig: 5 2 x - 1 · 9 x - 2 15 2 x - 3

  1. = 5 2 x - 1 · ( 3 2 ) x - 2 ( 5 . 3 ) 2 x - 3 = 5 2 x - 1 · 3 2 x - 4 5 2 x - 3 · 3 2 x - 3
  2. = 5 2 x - 1 - 2 x + 3 · 3 2 x - 4 - 2 x + 3 = 5 2 · 3 - 1
  3. = 25 3

Ondersoek: eksponensiale

Skryf die korrekte antwoord in the Antwoord kolom. Die beskikbare antwoorde is: 3 2 , 1, - 1 , - 1 3 , 8. Antwoorde mag herhaal word.

Vraag Antwoord
2 3
7 3 - 3
( 2 3 ) - 1
8 7 - 6
( - 3 ) - 1
( - 1 ) 23

Die volgende video gee 'n voorbeeld van hoe om sommige van die konsepte wat in hierdie hoofstuk gedek is, te gebruik.

Khan academy video oor eksponente 5 (in engels)

Hoofstukoefeninge

  1. Vereenvoudig so ver as moontlik.
    1. 302 0
    2. 1 0
    3. ( x y z ) 0
    4. [ ( 3 x 4 y 7 z 12 ) 5 ( - 5 x 9 y 3 z 4 ) 2 ] 0
    5. ( 2 x ) 3
    6. ( - 2 x ) 3
    7. ( 2 x ) 4
    8. ( - 2 x ) 4

  2. Vereenvoudig sonder om 'n sakrekenaar te gebruik. Skryf antwoorde met positiewe eksponente.
    1. 3 x - 3 ( 3 x ) 2
    2. 5 x 0 + 8 - 2 - ( 1 2 ) - 2 · 1 x
    3. 5 b - 3 5 b + 1

  3. Vereenvoudig en wys alle stappe.
    1. 2 a - 2 . 3 a + 3 6 a
    2. a 2 m + n + p a m + n + p · a m
    3. 3 n · 9 n - 3 27 n - 1
    4. ( 2 x 2 a y - b ) 3
    5. 2 3 x - 1 · 8 x + 1 4 2 x - 2
    6. 6 2 x · 11 2 x 22 2 x - 1 · 3 2 x

  4. Vereenvoudig sonder om 'n sakrekenaar te gebruik.
    1. ( - 3 ) - 3 · ( - 3 ) 2 ( - 3 ) - 4
    2. ( 3 - 1 + 2 - 1 ) - 1
    3. 9 n - 1 · 27 3 - 2 n 81 2 - n
    4. 2 3 n + 2 · 8 n - 3 4 3 n - 2

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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
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Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
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
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 10) [caps]. OpenStax CNX. Aug 04, 2011 Download for free at http://cnx.org/content/col11328/1.4
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