Noen nyttige potensregler:
a 0 = 1 a 1 = a a m ⋅ a n = a m + n ( a m ) n = a m ⋅ n ( a ⋅ b ) n = a n ⋅ b n \begin{aligned}
a^0 &= 1 \\
a^1 &= a \\
a^m \cdot a^n &= a^{m + n} \\
(a^m)^n & = a^{m \cdot n} \\
(a\cdot b)^n & = a^n \cdot b^n
\end{aligned} a 0 a 1 a m ⋅ a n ( a m ) n ( a ⋅ b ) n = 1 = a = a m + n = a m ⋅ n = a n ⋅ b n
a − n = 1 a n a m a n = a m − n a 1 / n = a n \begin{aligned}
a^{-n} & = \frac{1}{a^n} \\
\frac{a^m}{a^n} & = a^{m-n} \\
a^{1/n} & = \sqrt[n]{a}
\end{aligned} a − n a n a m a 1/ n = a n 1 = a m − n = n a
der .
+ Grunntallet
Et vanlige grunntall er :
Eulers tall, , er en konstant som er oppkalt etter en sveitsisk fyr, Leonhard Euler som levde på 1700-tallet. Denne konstanten dukker opp i mange forskjellige grener av matematikk og fysikk. Og derfor var det like greit at den fikk en egen bokstav (akkurat som ).
En av de mer vanlige definisjonene på :
e = lim x → ∞ ( 1 + 1 x ) x = 2.718281828 … e = \lim_{x \to \infty} \left(1 + \frac{1}{x} \right)^x = 2.718281828… e = x → ∞ lim ( 1 + x 1 ) x = 2.718281828 …
+ Grunntallet 10
Et vanlige grunntall er 10:
Veldig store og veldig små tall skrives ofte på det vi kaller vitenskapelig form. Da bruker vi grunntallet 10. For eksempel:
12000000 = 1.2 ⋅ 1 0 7 0.00000012 = 1.2 ⋅ 1 0 − 7 12000000 = 1.2 \cdot 10^7 \\
0.00000012 = 1.2 \cdot 10^{-7}
12000000 = 1.2 ⋅ 1 0 7 0.00000012 = 1.2 ⋅ 1 0 − 7 Eksponenten forteller hvor mange plasser vi må flytte kommaet.
+ Eksempel 1:
Uten formel:
a 2 ⋅ a 3 = a ⋅ a ⋅ a ⋅ a ⋅ a = a 5 \textcolor{blue}{a^2} \cdot \textcolor{red}{a^3} = \textcolor{blue}{a \cdot a} \cdot \textcolor{red}{a \cdot a \cdot a} = a^5 a 2 ⋅ a 3 = a ⋅ a ⋅ a ⋅ a ⋅ a = a 5 Med formel:
a 2 ⋅ a 3 = a 2 + 3 = a 5 \textcolor{blue}{a^2} \cdot \textcolor{red}{a^3} = a^{\textcolor{blue}{2} + \textcolor{red}{3}} = a^5 a 2 ⋅ a 3 = a 2 + 3 = a 5 Og, vips, er vi ferdige.
Eksempel: .
+ Eksempel 2:
Uten formel:
a 7 a 3 = a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a a ⋅ a ⋅ a = a ⋅ a ⋅ a ⋅ a = a 4 \frac{\textcolor{blue}{a^7}}{\textcolor{red}{a^3}} = \frac{\cancel{\textcolor{blue}{a \cdot a \cdot a}} \textcolor{blue}{\cdot a \cdot a \cdot a \cdot a}}{\cancel{\textcolor{red}{a \cdot a \cdot a}}} = \textcolor{blue}{a \cdot a \cdot a \cdot a} = a^4 a 3 a 7 = a ⋅ a ⋅ a a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a = a ⋅ a ⋅ a ⋅ a = a 4 Med formel:
a 7 a 3 = a 7 − 3 = a 4 \frac{\textcolor{blue}{a^7}}{\textcolor{red}{a^3}} = a^{\textcolor{blue}{7} - \textcolor{red}{3}} = a^4 a 3 a 7 = a 7 − 3 = a 4 Og, vips, er vi ferdige.
Eksempel: .
+ Eksempel 3:
Uten formel:
( a 4 ) 3 = a 4 ⋅ a 4 ⋅ a 4 = a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a = a 12 (a^4)^3
= \textcolor{red}{a^4} \cdot \textcolor{blue}{a^4} \cdot \textcolor{green}{a^4}
= \textcolor{red}{a \cdot a \cdot a \cdot a} \cdot \textcolor{blue}{a \cdot a \cdot a \cdot a} \cdot \textcolor{green}{a \cdot a \cdot a \cdot a}
= a^{12} ( a 4 ) 3 = a 4 ⋅ a 4 ⋅ a 4 = a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a ⋅ a = a 12 Med formel:
( a 4 ) 3 = a 4 ⋅ 3 = a 12 (a^{\textcolor{red}{4}})^{\textcolor{blue}{3}} = a^{\textcolor{red}{4} \cdot \textcolor{blue}{3}} = a^{12} ( a 4 ) 3 = a 4 ⋅ 3 = a 12 Og, vips, er vi ferdige.
Eksempel: .
+ Eksempel 4:
Uten formel:
a 4 ⋅ b 2 = a ⋅ a ⋅ a ⋅ a ⋅ b ⋅ b = ( a ⋅ a ⋅ b ) 2 = a ⋅ a ⋅ b = a 2 b \sqrt{\textcolor{red}{a^4} \cdot \textcolor{blue}{b^2}}
= \sqrt{\textcolor{red}{a \cdot a \cdot a \cdot a} \cdot \textcolor{blue}{b \cdot b}}
= \sqrt{(\textcolor{red}{a \cdot a} \cdot \textcolor{blue}{b})^2}
= \textcolor{red}{a \cdot a} \cdot \textcolor{blue}{b} = a^2 b a 4 ⋅ b 2 = a ⋅ a ⋅ a ⋅ a ⋅ b ⋅ b = ( a ⋅ a ⋅ b ) 2 = a ⋅ a ⋅ b = a 2 b Med formel:
a 4 b 2 = ( a 4 b 2 ) 1 2 = a 4 ⋅ 1 2 b 2 ⋅ 1 2 = a 2 b 1 \sqrt{a^{\textcolor{red}{4}} b^{\textcolor{blue}{2}}}
= (a^{\textcolor{red}{4}} b^{\textcolor{blue}{2}} )^{\frac{1}{2}}
= a^{\textcolor{red}{4} \cdot \frac{1}{2}} b^{\textcolor{blue}{2} \cdot \frac{1}{2}}
= a^{\textcolor{red}{2}} b^{\textcolor{blue}{1}} a 4 b 2 = ( a 4 b 2 ) 2 1 = a 4 ⋅ 2 1 b 2 ⋅ 2 1 = a 2 b 1 Og, vips, er vi ferdige.
Eksempel: .
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