### Teorija

Par eksponentvienādojumu sauc tādu vienādojumu, kur nezināmais atrodas kāpinātājā. Piemēram, ${3}^{x}=27,\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}{6}^{x-3}-{6}^{3}=36$
Eksponentvienādojumu risināšanā nepieciešams izmantot pakāpju īpašības un definīcijas.
Par reāla skaitļa $a$ pakāpi ar naturālu kāpinātāju $n$ sauc reizinājumu, kurā skaitlis $a$ ņemts $n$ reizes.
$\begin{array}{l}{a}^{n}=\underset{⏟}{a\cdot a\cdot a\cdot ...\cdot a}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}{a}^{1}=a,\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}{a}^{0}=1\phantom{\rule{0.147em}{0ex}}\left(\mathrm{ja}\phantom{\rule{0.147em}{0ex}}a\ne 0\right)\\ \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}n\phantom{\rule{0.147em}{0ex}}\mathrm{reizes}\end{array}$
Piemērs:
$\begin{array}{l}{4}^{3}=4\cdot 4\cdot 4=64\\ {\left(-3\right)}^{4}=\left(-3\right)\left(-3\right)\left(-3\right)\left(-3\right)=81\end{array}$

$\begin{array}{l}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\text{kāpinātājs}\\ \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}↓\\ \text{bāze}\phantom{\rule{0.147em}{0ex}}\to {2}^{3}=8\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}↖↗\\ \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\text{pakāpe}\end{array}$

Ja negatīva skaitļa kāpinātājs ir pāra skaitlis, tad skaitļa pakāpe ir pozitīvs skaitlis.
Ja negatīva skaitļa kāpinātājs ir nepāra skaitlis, tad pakāpe ir negatīvs skaitlis.
Piemērs:
${\left(-2\right)}^{4}=16;\phantom{\rule{1.029em}{0ex}}{\left(-2\right)}^{3}=-8$

Ja kāpinātājs ir vesels negatīvs skaitlis:
${a}^{-n}=\frac{1}{{a}^{n}}$

Piemērs:
Pārveido par pakāpi!

$\begin{array}{l}\frac{1}{8}=\frac{1}{{2}^{3}}={2}^{-3}\\ \phantom{\rule{0.147em}{0ex}}\\ \frac{1}{{x}^{-4}}={x}^{4}\end{array}$
Ja $a>0$ un $m$, $n$ ir naturāli skaitļi, tad ${a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}}$
Piemērs:
Pārveido par pakāpi!

$\begin{array}{l}\sqrt[3]{{x}^{5}}={x}^{\frac{5}{3}}\\ \sqrt{2}\cdot {2}^{3}={2}^{\frac{1}{2}}\cdot {2}^{3}={2}^{3\frac{1}{2}}\end{array}$

Kāpināšanas īpašības
$\begin{array}{l}1\right)\phantom{\rule{0.147em}{0ex}}{a}^{m}\cdot {a}^{n}={a}^{m+n}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ 2\right)\phantom{\rule{0.147em}{0ex}}{a}^{m}\cdot {b}^{m}={\left(\mathit{ab}\right)}^{m}\\ 3\right)\phantom{\rule{0.147em}{0ex}}{a}^{m}:{a}^{n}={a}^{m-n}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\\ 4\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\frac{{a}^{n}}{{b}^{n}}={\left(\frac{a}{b}\right)}^{n}\\ 5\right)\phantom{\rule{0.147em}{0ex}}{\left({a}^{m}\right)}^{n}={a}^{\mathit{mn}}\\ 6\right)\phantom{\rule{0.147em}{0ex}}{\left(\frac{a}{b}\right)}^{-n}={\left(\frac{b}{a}\right)}^{n}\end{array}$

1. pakāpju reizināšana, ja bāzes ir vienādas;
2. reizinājuma kāpināšana;
3. pakāpju dalīšana, ja bāzes ir vienādas;
4. dalījuma kāpināšana;
5. pakāpes kāpināšana;
6. dalījuma pakāpe ar negatīvu kāpinātāju.

$\begin{array}{l}1\right)\phantom{\rule{0.147em}{0ex}}{x}^{3}\cdot {x}^{6}={x}^{3+6}\phantom{\rule{0.147em}{0ex}}={x}^{9}\\ 2\right)\phantom{\rule{0.147em}{0ex}}{2}^{3}\cdot {5}^{3}={10}^{3}=1000\\ 3\right)\phantom{\rule{0.147em}{0ex}}\frac{{3}^{8}}{{3}^{6}}={3}^{8}:{3}^{6}={3}^{8-6}={3}^{2}=9\\ 4\right)\phantom{\rule{0.147em}{0ex}}\frac{{8}^{6}}{{4}^{6}}={\left(\frac{8}{4}\right)}^{6}={2}^{6}=64\\ 5\right)\phantom{\rule{0.147em}{0ex}}{\left({y}^{3}\right)}^{4}={y}^{3\cdot 4}={y}^{12}\\ 6\right)\phantom{\rule{0.147em}{0ex}}{\left(\frac{2}{3}\right)}^{-4}={\left(\frac{3}{2}\right)}^{4}\end{array}$