Definisjoner

$a^{n} \overset{def}{=} \underset{n ganger}{\underset{︸}{a \cdot a \cdot a \cdot . . . \cdot a}} a^{- n} \overset{def}{=} \frac{1}{a^{n}} a^{0} \overset{def}{=} 1$

Regneregler

$a^{m} \cdot a^{n} = a^{m + n} \frac{a^{m}}{a^{n}} = a^{m - n}$

${(a \cdot b)}^{n} = a^{n} \cdot b^{n} {(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}} {(a^{n})}^{m} = a^{m \cdot n}$

1.1 Bruk potensreglene og regn ut

a) $4^{2} \cdot 4^{5}$

vis fasit

$= 4^{2 + 5} = 4^{7}$

b) $3^{4} \cdot 3$

vis fasit

$= 3^{4 + 1} = 3^{5}$

c) $\frac{3^{6}}{3^{3}}$

vis fasit

$= 3^{6 - 3} = 3^{3}$

d) $3^{2} \cdot 2^{2}$

vis fasit

$= 9 \cdot 4 = 36$

e) $\frac{4^{2}}{3}$

vis fasit

$= \frac{16}{3}$

f) ${(3^{2})}^{2}$

vis fasit

$= 3^{2 \cdot 2} = 3^{4}$

g) ${(\frac{4^{2}}{3})}^{2}$

vis fasit

$= \frac{4^{2 \cdot 2}}{3^{2}} = \frac{256}{9}$

h) ${(\frac{4}{3})}^{2}$

vis fasit

$= \frac{4^{2}}{3^{2}} = \frac{16}{9}$

1.2 Bruk potensreglene og regn ut

a) $x^{3} \cdot x^{2}$

vis fasit

$= x^{3 + 2} = x^{5}$

b) $\frac{x^{4}}{x^{2}}$

vis fasit

$= x^{4 - 2} = x^{2}$

c) $\frac{b^{3} \cdot b^{2}}{b^{4}}$

vis fasit

$= \frac{b^{3 + 2}}{b^{4}} = b^{5 - 4} = b^{1} = b$

d) $\frac{y^{3} \cdot y^{3}}{y^{3} \cdot y}$

vis fasit

$= \frac{y^{3 + 3}}{y^{3 + 1}} = \frac{y^{6}}{y^{4}} = y^{6 - 4} = y^{2}$

e) ${(a b)}^{2} \cdot a^{2}$

vis fasit

$= a^{2} \cdot b^{2} \cdot a^{2} = a^{2 + 2} \cdot b^{2} = a^{4} b^{2}$

f) $\frac{{(x y)}^{4}}{y^{3}}$

vis fasit

$= \frac{x^{4} \cdot y^{4}}{y^{3}} = x^{4} \cdot y^{4 - 3} = x^{4} \cdot y^{1} = x^{4} y$

g) $\frac{{(a^{2} b^{3})}^{3}}{a^{5} b^{8}}$

vis fasit

$= \frac{a^{2 \cdot 3} b^{3 \cdot 3}}{a^{5} b^{8}} = \frac{a^{6} b^{9}}{a^{5} b^{8}} = a^{6 - 5} b^{9 - 8} = a b$

h) ${(2 x^{2} y^{3})}^{2} \cdot {(2 x y)}^{3}$

vis fasit

$= 2^{2} x^{2 \cdot 2} y^{3 \cdot 2} \cdot 2^{3} x^{3} y^{3} = 4 x^{4} y^{6} \cdot 8 x^{3} y^{3} = 4 \cdot 8 x^{4 + 3} y^{6 + 3} = 32 x^{7} y^{9}$

1.3 Bruk potensreglene og regn ut

a) ${(2 a)}^{2}$

vis fasit

$= 2^{2} \cdot a^{2} = 4 a^{2}$

b) ${(3 a b)}^{3}$

vis fasit

$= 3^{3} \cdot a^{3} \cdot b^{3} = 27 a^{3} b^{3}$

c) $\frac{{(2 x)}^{3}}{8 x}$

vis fasit

$= \frac{2^{3} \cdot x^{3}}{2^{3} \cdot x} = 2^{3 - 3} \cdot x^{3 - 1} = 2^{0} \cdot x^{2} = x^{2}$

d) $\frac{{(3 x \cdot 2 y)}^{2}}{x y}$

vis fasit

$= \frac{3^{2} \cdot x^{2} \cdot 2^{2} \cdot y^{2}}{x \cdot y} = 9 \cdot 4 \cdot x^{2 - 1} \cdot y^{2 - 1} = 36 x y$

1.4 Bruk potensreglene og regn ut

a) $\frac{(2 x)^{2}}{4 x^{2}}$

vis fasit

$= \frac{2^{2} \cdot x^{2}}{4 \cdot x^{2}} = \frac{4 \cdot x^{2}}{4 \cdot x^{2}} = 1$

b) $\frac{(a b)^{3}}{a^{3} b^{- 2}}$

vis fasit

$= \frac{a^{3} \cdot b^{3}}{a^{3} \cdot b^{- 2}} = a^{3 - 3} \cdot b^{3 + 2} = a^{0} \cdot b^{5} = 1 \cdot b^{5} = b^{5}$

c) $\frac{a^{2} \cdot b^{- 2} \cdot a^{- 3}}{a^{2} \cdot a \cdot b^{- 3}}$

vis fasit

$= a^{2 - 3 - 2 - 1} \cdot b^{- 2 + 3} = a^{- 4} \cdot b^{1} = \frac{b}{a^{4}}$

d) $(- 2)^{2} \cdot (- 2)^{3}$

vis fasit

$= 4 \cdot (- 8) = - 32$

e) Kontroller svarene dine med CAS i Geogebra.

1.5 Regn ut og skriv svaret med positiv eksponent

a) $2^{- 2}$

vis fasit

$= \frac{2}{2^{2}} = \frac{1}{4}$

b) ${(2^{- 2})}^{3}$

vis fasit

$= 2^{- 2 \cdot 3} = 2^{- 6} = \frac{1}{2^{6}} = \frac{1}{64}$

c) $x^{4} \cdot x^{- 5}$

vis fasit

$= x^{4 - 5} = x^{- 1} = \frac{1}{x}$

d) $\frac{y^{3}}{y^{5}}$

vis fasit

$= y^{3 - 5} = y^{- 2} = \frac{1}{y^{2}}$

1.6 Bruk potensreglene og regn ut

a) $(- a^{2})^{3}$

vis fasit

$= {((- 1) \cdot a^{2})}^{3} = {(- 1)}^{3} \cdot a^{2 \cdot 3} = - 1 \cdot a^{6} = - a^{6}$

b) $y^{- 3} \cdot {(y^{2})}^{4} \cdot y^{0}$

vis fasit

$= y^{- 3} \cdot y^{2 \cdot 4} \cdot 1 = y^{- 3 + 8} = y^{5}$

c) $\frac{{(b^{3})}^{- 2} \cdot a^{3}}{b \cdot {(b^{2})}^{2}}$

vis fasit

$= \frac{b^{- 6} \cdot a^{3}}{b \cdot b^{4}} = a^{3} \cdot b^{- 6 - 1 - 4} = a^{3} \cdot b^{- 11} = \frac{a^{3}}{b^{11}}$

d) $\frac{x^{2} \cdot y^{- 2} \cdot z^{3}}{{(x \cdot y \cdot z)}^{2}}$

vis fasit

$= \frac{x^{2} \cdot y^{- 2} \cdot z^{3}}{x^{2} \cdot y^{2} \cdot z^{2}} = x^{2 - 2} \cdot y^{- 2 - 2} \cdot z^{3 - 2} = x^{0} \cdot y^{- 4} \cdot z^{1} = \frac{z}{y^{4}}$