# Oppgaver: Potenser :::{margin} Potensregel 1 $$ x^a \cdot x^b = x^{a+b} $$ ::: :::::::::::::::{exercise} Oppgave 1 --- level: 1 --- Bruk potensreglene til å skrive så enkelt som mulig. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ x^2 \cdot x^3 $$ ::::{answer} $$ x^{2+3} = x^5 $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ x^4 \cdot x^2 $$ ::::{answer} $$ x^{4+2} = x^6 $$ :::: ::::::::::::: :::::::::::::{tab-item} c :::{sidebar} Bruk potensregel 1 flere ganger! ::: $$ x^2 \cdot x^7 \cdot x^3 $$ ::::{answer} $$ x^{2+7+3} = x^{12} $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ x^{10} \cdot x^2 \cdot x^4 $$ ::::{answer} $$ x^{10+2+4} = x^{16} $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::{margin} Potensregel 2 $$ \dfrac{x^a}{x^b} = x^{a-b} $$ ::: :::::::::::::::{exercise} Oppgave 2 --- level: 1 --- Bruk potensreglene til å skrive så enkelt som mulig. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ \dfrac{x^5}{x^2} $$ ::::{answer} $$ x^{5-2} = x^3 $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ \dfrac{x^4}{x^2} $$ ::::{answer} $$ x^{4-2} = x^2 $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ \dfrac{x^3 \cdot x^6}{x^2} $$ ::::{answer} $$ x^{3+6-2} = x^{7} $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ \dfrac{x^2 \cdot x^9}{x^3 \cdot x^7} $$ ::::{answer} $$ x^{2+9-3-7} = x^{1} $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::{margin} Potensregel 3 $$ (x^a)^b = x^{a \cdot b} $$ ::: :::::::::::::::{exercise} Oppgave 3 --- level: 1 --- Bruk potensreglene til å skrive så enkelt som mulig. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ (x^2)^3 $$ ::::{answer} $$ x^{2 \cdot 3} = x^6 $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ (x^3)^2 $$ ::::{answer} $$ x^{3 \cdot 2} = x^6 $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ (x^4)^2 $$ ::::{answer} $$ x^{4 \cdot 2} = x^8 $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ (x^5)^3 $$ ::::{answer} $$ x^{5 \cdot 3} = x^{15} $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::{margin} Potensregel 4 $$ (x \cdot y)^a = x^a \cdot y^a $$ ::: :::::::::::::::{exercise} Oppgave 4 --- level: 1 --- Bruk potensregler til å skrive så enkelt som mulig. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ (2\cdot x)^3 $$ ::::{answer} $$ (2\cdot x)^3 = 2^3 \cdot x^3 = 8x^3 $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ (3x)^2 $$ ::::{answer} $$ (3x)^2 = 3^2 \cdot x^2 = 9x^2 $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ (4y)^3 $$ ::::{answer} $$ (4y)^3 = 4^3 \cdot y^3 = 64y^3 $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ (2xy)^4 $$ ::::{answer} $$ (2xy)^4 = 2^4 \cdot x^4 \cdot y^4 = 16x^4y^4 $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::{margin} Potensregel 5 $$ x^0 = 1 \qfor x \neq 0 $$ ::: :::::::::::::::{exercise} Oppgave 5 --- level: 1 --- Bruk potensregler til å skrive så enkelt som mulig. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ 10^0 $$ ::::{answer} $$ 10^0 = 1 $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ 72^0 $$ ::::{answer} $$ 72^0 = 1 $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ 1000^0 $$ ::::{answer} $$ 1000^0 = 1 $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::{margin} Potensregel 6 $$ x^{-a} = \dfrac{1}{x^a} $$ ::: :::::::::::::::{exercise} Oppgave 6 --- level: 1 --- Bruk potensreglene til å skriv så enkelt som mulig. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ 2^{-3} $$ ::::{answer} $$ 2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8} $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ x^{-2} $$ ::::{answer} $$ x^{-2} = \dfrac{1}{x^2} $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ x^2\cdot x^{-5} $$ ::::{answer} $$ x^2\cdot x^{-5} = x^{2-5} = x^{-3} = \dfrac{1}{x^3} $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ x^{-5} \cdot x^4 $$ ::::{answer} $$ x^{-5} \cdot x^4 = x^{-5+4} = x^{-1} = \dfrac{1}{x} $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 7 --- level: 2 --- Bruk potensreglene til å skrive så enkelt som mulig. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ (2x)^3 \cdot (3x^2) $$ ::::{answer} $$ (2x)^3 \cdot (3x^2) = 2^3 \cdot x^3 \cdot 3 \cdot x^2 = 8 \cdot 3 \cdot x^{3+2} = 24x^5 $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ \dfrac{(3x^2)^3}{(2x)^4} $$ ::::{answer} $$ \dfrac{(3x^2)^3}{(2x)^4} = \dfrac{27x^6}{16x^4} = \dfrac{27}{16}x^{6-4} = \dfrac{27}{16}x^2 $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ (x^{-1})^2 \cdot (2x)^3 $$ ::::{answer} $$ (x^{-1})^2 \cdot (2x)^3 = x^{-2} \cdot 2^3 \cdot x^3 = 8x^{3-2} = 8x $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ (2x^4 \cdot 3y^3)^2 $$ ::::{answer} $$ (2x^4 \cdot 3y^3)^2 = (2^2 \cdot x^{4 \cdot 2} \cdot 3^2 \cdot y^{3 \cdot 2}) = 4x^8 \cdot 9y^6 = 36x^8y^6 $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 8 --- level: 2 --- Bruk potensreglene til å skrive så enkelt som mulig. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ \dfrac{(2x^2y)^3}{(3y)^{-2}} $$ ::::{answer} $$ \dfrac{(2x^2y)^3}{(3y)^{-2}} = \dfrac{8x^6y^3}{\dfrac{1}{9y^2}} = 8x^6y^3 \cdot 9y^2 = 72x^6y^5 $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ \dfrac{(x^2y^3)^{-1}}{(4x^3y^2)^{-2}} $$ ::::{answer} $$ \dfrac{(x^2y^3)^{-1}}{(4x^3y^2)^{-2}} = \dfrac{1}{x^2y^3} \cdot \dfrac{16x^6y^4}{1} = 16x^{6-2}y^{4-3} = 16x^4y $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ \dfrac{(a^2 b^3)^4 b^{-2}}{(2a^3)^2 a^{-3}} $$ ::::{answer} $$ \dfrac{(a^2 b^3)^4 b^{-2}}{(2a^3)^2 a^{-3}} = \dfrac{(a^8 b^{12})b^{-2}}{4a^6 a^{-3}} = \dfrac{a^8 b^{12-2}}{4a^{6-3}} = \dfrac{a^8 b^{10}}{4a^3} = \dfrac{a^{8-3}b^{10}}{4} = \dfrac{a^5b^{10}}{4} $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ \left(\dfrac{a^2}{b^3}\right)^4 \cdot \dfrac{b^4}{(2a^{-3})^2} $$ ::::{answer} $$ \left(\dfrac{a^2}{b^3}\right)^4 \cdot \dfrac{b^4}{(2a^{-3})^2} = \dfrac{a^8}{b^{12}} \cdot \dfrac{b^4}{4a^{-6}} = \dfrac{a^8 b^{4-12}}{4a^{-6}} = \dfrac{a^8b^{-8}}{4a^{-6}} = \dfrac{a^{8-(-6)}b^{-8}}{4} = \dfrac{a^{14}}{4b^8} $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::{margin} Potensregel for $n$-te røtter $$ \sqrt[n]{x} = x^\tfrac{1}{n} $$ ::: :::::::::::::::{exercise} Oppgave 9 --- level: 2 --- ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Regn ut. $$ 4^{\tfrac{1}{2}} $$ ::::{answer} $$ 4^{\tfrac{1}{2}} = \sqrt{4} = 2 $$ :::: ::::::::::::: :::::::::::::{tab-item} b Regn ut. $$ 8^{\tfrac{1}{3}} $$ ::::{answer} $$ 8^{\tfrac{1}{3}} = \sqrt[3]{8} = 2 $$ :::: ::::::::::::: :::::::::::::{tab-item} c Regn ut. $$ 16^{-\tfrac{1}{4}} $$ ::::{answer} $$ 16^{-\tfrac{1}{4}} = \dfrac{1}{16^{\tfrac{1}{4}}} = \dfrac{1}{\sqrt[4]{16}} = \dfrac{1}{2} = 2^{-1} $$ :::: ::::::::::::: :::::::::::::{tab-item} d Skriv så enkelt som mulig. $$ \sqrt[4]{x^8} $$ ::::{answer} $$ \sqrt[4]{x^8} = (x^8)^{\tfrac{1}{4}} = x^{8 \cdot \tfrac{1}{4}} = x^2 $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 10 --- level: 2 --- Bruk potensreglene til å skrive så enkelt som mulig. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ \sqrt[3]{x^4} \cdot x^\tfrac{2}{3} $$ ::::{answer} $$ \sqrt[3]{x^4} \cdot x^\tfrac{2}{3} = (x^4)^{\tfrac{1}{3}} \cdot x^{2/3} = x^{4/3} \cdot x^{2/3} = x^{4/3 + 2/3} = x^{6/3} = x^2 $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ (x^\tfrac{3}{4} \cdot \sqrt[4]{x})^2 $$ ::::{answer} $$ (x^\tfrac{3}{4} \cdot \sqrt[4]{x})^2 = (x^\tfrac{3}{4} \cdot (x^{1/4}))^2 = (x^{3/4 + 1/4})^2 = (x^{4/4})^2 = (x^1)^2 = x^2 $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ \dfrac{\sqrt[4]{x^3} \cdot x^{5/4}}{\sqrt{x}} $$ ::::{answer} $$ \dfrac{\sqrt[4]{x^3} \cdot x^{5/4}}{\sqrt{x}} = \dfrac{x^{3/4} \cdot x^{5/4}}{x^{1/2}} = \dfrac{x^{2}}{x^{1/2}} = x^{2-1/2} = x^{3/2} $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 11 --- level: 3 --- ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a Skriv om til en så enkel som mulig potens med grunntall $2$. $$ \dfrac{2^0 \cdot 2^3 \cdot 2^4 \cdot (2^3)^2 \cdot 4^{-2}}{2^2 \cdot 4} \cdot 2^{-3} $$ ::::{answer} $$ 2^2 $$ :::: ::::::::::::: :::::::::::::{tab-item} b Skriv om til et produkt av potenser der grunntallene er primtall. $$ \dfrac{9(3^2 + 27^{2/3})^2}{81 \sqrt[3]{3^2}} $$ ::::{answer} $$ 2^2 \cdot 3^{4/3} $$ :::: ::::::::::::: :::::::::::::{tab-item} c Skriv så enkelt som mulig. $$ \dfrac{(2x)^{-2}y^{-3}(x^2y + xy^2)}{2^{-2}x^{-1}y^{-2}} $$ ::::{answer} $$ x + y $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::{margin} Tips: Oppgave 12, 13 og 14 Uttrykkene her er ganske sammensatte. Det er lurt å prøve å bruke én og én potensregel av gangen på hver del av uttrykket for å senke sjansen for slurvefeil. Hold tunga rett i munn og ta deg god tid! ::: :::::::::::::::{exercise} Oppgave 12 --- level: 3 --- Skriv så enkelt som mulig. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ (2y^3)^2\cdot \sqrt[4]{y^{-2}} $$ ::::{answer} $$ 4y^{11/2} $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ \dfrac{(3x^3)^2 \cdot \sqrt[5]{x^{10}}}{(\sqrt[3]{x})^6 \cdot (3x)^{-2}} $$ ::::{answer} $$ 81x^{8} $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ \left(\dfrac{x^{-2}}{5}\right)^{-1} \cdot \dfrac{(2x^3)^{-1}}{20x^2 \cdot (5x^{-1})^2} $$ ::::{answer} $$ \dfrac{1}{200x} $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ (3a^2b^5)^{-2} \cdot 3ab^{-1} $$ ::::{answer} $$ \dfrac{1}{3a^3b^{11}} $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 13 --- level: 3 --- Skriv så enkelt som mulig. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ \dfrac{x^{-2}a^5 (2x^{-4})^2}{(3x^2a^{-2})^{-1}} $$ ::::{answer} $$ \dfrac{12a^3}{x^8} $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ \left(\dfrac{(8x^{-2})^{-1}}{2^{-1}x^3}\right)^{-1} $$ ::::{answer} $$ 4x $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ \dfrac{(a^3 b^2 c^{-1})^2 (\sqrt{a} \, b^2 c^{1/4})^8}{(ab^2)^2 (b^2 c)^3 (ac^2)^{-4}} $$ ::::{answer} $$ a^{12}b^{10}c^{5} $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ \dfrac{ (x^{-2} y^3 \sqrt{z})^4 \, (xz^{-1})^{-3}}{ (\sqrt[3]{x} \, y^{-1} z^2)^6 \, (y^2 z^{-3})^{-2} } $$ ::::{answer} $$ \dfrac{y^{22}}{x^{13}z^{13}} $$ :::: ::::::::::::: :::::::::::::: ::::::::::::::: --- :::::::::::::::{exercise} Oppgave 14 --- level: 3 --- Skriv så enkelt som mulig. ::::::::::::::{tab-set} --- class: tabs-parts --- :::::::::::::{tab-item} a $$ \dfrac{12x^2 \cdot (6x)^{-2}}{(3x^2)^{-3}} $$ ::::{answer} $$ 3x^5 $$ :::: ::::::::::::: :::::::::::::{tab-item} b $$ \dfrac{(8x^{-2})^{-1}}{(4x^3)^2 \cdot (2^{-1}x^3)^4} $$ ::::{answer} $$ \dfrac{1}{8x^{16}} $$ :::: ::::::::::::: :::::::::::::{tab-item} c $$ \dfrac{\left( (\sqrt[3]{2})^2 x \right)^3}{\sqrt[3]{x^2} \cdot \sqrt{\sqrt[6]{x}}} $$ ::::{answer} $$ 4x^2 $$ :::: ::::::::::::: :::::::::::::{tab-item} d $$ \dfrac{ \sqrt[5]{x^2} \cdot (5x^3)^2 \cdot (5x^4)^{-1} }{(\sqrt{5}x)^6 \cdot \sqrt[10]{x^4} \cdot (5^{-1}x^{-2})^2 } $$ ::::{answer} $$ 1 $$ :::: ::::::::::::: :::::::::::::: :::::::::::::::