Exponente

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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^{\frac{m}{n}} = \sqrt[n]{a^{m}}

Eksponensiële wet 7: $a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$

Ons sê dat $x$ is 'n $n$ de wortel van $b$ as $x^{n} = b$ en ons kan skryf $x = \sqrt[n]{b}$ . $n^{de}$ wortels geskryf met die radikale simbool, $\sqrt{}$ , 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^{\frac{m}{n}})}^{n} = a^{\frac{m}{n} \times n} = a^{m}

dus $a^{\frac{m}{n}}$ moet 'n $n$ de wortel van $a^{m}$ wees. Ons kan dus sê,

a^{\frac{m}{n}} = \sqrt[n]{a^{m}}

Byvoorbeeld,

2^{\frac{2}{3}} = \sqrt[3]{2^{2}}

'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 $x^{2} \geq 0$ 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:

{(\frac{5}{4^{- 1} - 9^{- 1}})}^{\frac{1}{2}}

Herskryf negatiewe eksponente as getalle met positiewe indukse
$\begin{matrix} = & {(\frac{5}{\frac{1}{4} - \frac{1}{9}})}^{\frac{1}{2}} \end{matrix}$
Vereenvoudig tussen hakies
$\begin{matrix} = & {(\frac{5}{1} \div \frac{9 - 4}{36})}^{\frac{1}{2}} \\ = & {(\frac{5}{1} \times \frac{36}{5})}^{\frac{1}{2}} \\ = & {(6^{2})}^{\frac{1}{2}} \end{matrix}$
Gebruik eksponensiële wet 6
$\begin{matrix} = & 6 \end{matrix}$

Vereenvoudig:

{(16 x^{4})}^{\frac{3}{4}}

Omskep die getal koëffisiënt na indeks-vorm met 'n prima basis
$\begin{matrix} = & {(2^{4} x^{4})}^{\frac{3}{4}} \end{matrix}$
Gebruik eksponensiële wette
$\begin{matrix} = & 2^{4 \times \frac{3}{4}} . x^{4 \times \frac{3}{4}} \\ = & 2^{3} . x^{3} \\ = & 8 x^{3} \end{matrix}$

Toepassing van wette

Gebruik al die wette om:

Vereenvoudig:

(a) $(x^{0}) + 5 x^{0} - {(0, 25)}^{- 0, 5} + 8^{\frac{2}{3}}$	(b) $s^{\frac{1}{2}} \div s^{\frac{1}{3}}$
(c) $\frac{12 m^{\frac{7}{9}}}{8 m^{- \frac{11}{9}}}$	(d) ${(64 m^{6})}^{\frac{2}{3}}$

Her-skryf die volgende uitdrukking as 'n krag van $x$ :
$x \sqrt{x \sqrt{x \sqrt{x \sqrt{x}}}}$

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?

$P o p u l a t i o n = I n i t i a l p o p u l a t i o n \times {(1 + g r o w t h p e r c e n t a g e)}^{t i m e p e r i o d i n h o u r s}$
Daarom, in hierdie geval:

$P o p u l a t i o n = 10 {(1, 8)}^{n}$ , waar $n$ = aantel ure
In 5 ure
$P o p u l a t i o n = 10 {(1, 8)}^{5} = 189$
In 1 dag = 24 ure
$P o p u l a t i o n = 10 {(1, 8)}^{24} = 13 382 588$
In 1 week = 168 ure
$P o p u l a t i o n = 10 {(1, 8)}^{168} = 7, 687 \times 10^{43}$

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?

$P o p u l a t i o n = I n i t i a l p o p u l a t i o n \times {(1 + g r o w t h p e r c e n t a g e)}^{t i m e p e r i o d i n m o n t h s}$
Daarom, in hierdie geval:

$P o p u l a t i o n = 821 {(1, 02)}^{n}$ , waar $n$ = aantal maande
'n half jaar = 6 mande
$P o p u l a t i o n = 821 {(1, 02)}^{6} = 925$
10 jaar = 120 mande
$P o p u l a t i o n = 821 {(1, 02)}^{120} = 8 838$
'n 100 jaar= 1 200 mande
$P o p u l a t i o n = 821 {(1, 02)}^{1 200} = 1, 716 \times 10^{13}$

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

Einde van hoofstuk oefeninge

Vereenvoudig so ver as moontlik:
1. $8^{- \frac{2}{3}}$
2. $\sqrt{16} + 8^{- \frac{2}{3}}$

Vereenvoudig:

(a) ${(x^{3})}^{\frac{4}{3}}$	(b) ${(s^{2})}^{\frac{1}{2}}$
(c) ${(m^{5})}^{\frac{5}{3}}$	(d) ${(- m^{2})}^{\frac{4}{3}}$
(e) $- {(m^{2})}^{\frac{4}{3}}$	(f) ${(3 y^{\frac{4}{3}})}^{4}$

Vereenvoudig so veel as wat jy kan:
$\frac{3 a^{- 2} b^{15} c^{- 5}}{{(a^{- 4}, b^{3}, c)}^{\frac{- 5}{2}}}$
Vereenvoudig so veel as wat jy kan:
${(9, a^{6}, b^{4})}^{\frac{1}{2}}$
Vereenvoudig so veel as wat jy kan:
${(a^{\frac{3}{2}}, b^{\frac{3}{4}})}^{16}$
Vereenvoudig:
$x^{3} \sqrt{x}$
Vereenvoudig:
$\sqrt[3]{x^{4} b^{5}}$
Herskryf die volgende uitdrukking as 'n krag van $x$ :
$\frac{x \sqrt{x \sqrt{x \sqrt{x \sqrt{x}}}}}{\sqrt[3]{x}}$

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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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Exponente

Eksponente - graad 11

Inleiding

Wette van eksponente

Eksponensiële wet 7: a m n = a m n

Komplekse getalle

Toepassing van wette

Eksponensiële in die werklike wêreld

Einde van hoofstuk oefeninge

Questions & Answers

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Eksponensiële wet 7: $a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$