1.1 Potenssi ja verrannollisuus

124 WIP

n=1,2,3,4...
$a_n=\left(\frac{n}{n+1}\right)^{1-n}$
$n=1$
$=\left(\frac{1}{1+1}\right)1-1=1$
$n=2$
$=\left(\frac{2}{2+1}\right)^{1-2}=\frac{3}{2}$
$n=3$
$=\left(\frac{3}{3+1}\right)^{1-3}=\frac{16}{4}=4$
$n=4$
$=\left(\frac{4}{4+1}\right)^{1-4}=\frac{125}{64}$

114

$10^{-10}mm^3=10^{-16}dm^3$

vesimolekyylien määrä maljakossa on

$3\cdot10^{25}\cdot10^{-16}dm^3=3\cdot10^9$

maljakkoon mahtuu 3 miljardia vesimolekyyliä

113

a)
0,00123
b)
$-5\cdot10^{-4}$
c)
537,34
d)
$7\cdot10^1+5\cdot10^0+2\cdot10^{-1}+3\cdot10^{-2}$

110

a)

$\left(\frac{5}{4}\right)^{-2}+5^{-1}=\frac{16}{25}+\frac{1}{5}=\frac{21}{25}$

b)

$\left(\frac{1}{2x}\right)^{-3}-\left(-2x\right)^3=8x^3+8x^3=16x^3$

c)

$\left(7x\right)^{11}\cdot\left(\frac{7}{x}\right)^{-9}=\frac{7^{11}x^{11}}{1}\cdot\frac{x^9}{7^9}=\frac{7^2x^{20}}{1}=49x^{20}$

105

a)
$2x^{-4}=\frac{2}{x^4}$
$\frac{3}{5}x^{-2}=\frac{3}{5x^2}$

$\left(\frac{5}{4x}\right)^{-2}=\frac{16x^2}{25}$

b)

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

$\frac{2}{3x^5}=\frac{2}{3}x^{-5}$

102

a)

$x^3\cdot x^8=x^{11}$

b)

$4x^7\cdot8x^{14}=32x^{21}$
c)

$-x^6\cdot5x^{-5}=-5x$

d)

$\left(3x\right)^2\cdot\left(-2x\right)^3=9x^2\cdot\left(-6x^3\right)=-54x^5$

101

a)
$\left(5x\right)^2=25x^2$

b)

$\left(x^2\right)^4=x^8$

c)

$\left(7x^3\right)^2=49x^6$

d)

$\left(-3x^5\right)^2=9x^{10}$

109

a)

$3^1+3^0+3^{-1}=3+1+\frac{1}{3}=4\ \frac{1}{3}$

b)

$4^{-2}+2^{-1}-2^1=\frac{1}{16}+\frac{1}{2}-2=\frac{9}{16}-2=-\frac{23}{16}$

c)

$\frac{1}{10}-10^{-2}+0{,}02=0{,}1-\frac{1}{100}+0{,}02=0{,}11$

d)

$-3^{-2}+\left(-6\right)^{-2}=-\frac{1}{9}+\frac{1}{36}=-\frac{4}{36}+\frac{1}{36}=-\frac{3}{36}=-\frac{1}{12}$

108

a)

$5^3\cdot2^3=10^3=1000$

b)

$\frac{18^4}{9^4}=\left(\frac{18}{9}\right)^4=2^4=16$

c)

$\left(\frac{1}{3}\right)^7\cdot3^6=3^{-7}\cdot3^6=3^{-1}=\frac{1}{3}$

d)

$2{,}5\cdot10^3\cdot4\cdot10^{-4}=10\cdot10^3\cdot10^{-4}=10^0=1$

118

a)
$s=kv^2$ , missä k on vakio
nopeudella 80 km/h jarrutusmatka on 30m, eli $30=k\cdot80^2$
$k=\frac{30}{80^2}=\frac{3}{640}$
$s=\frac{3}{640}v^2$
b)

Lasketaan jarrutusmatka nopeuden ollessa 100 km/h

$s=\frac{3}{640}\cdot100^2=46{,}875m\approx47m$

nopeudella 80 km/h jarrutusmatkan pituus on 30m

matka vähenee $47m-30m=17m$

lasketaan jarrutusmatka nopeuden ollessa 50km/h

$s=\frac{3}{640}\cdot50^2=11{,}71875\approx12m$

lasketaan jarrutusmatka nopeuden ollessa 30km/h

$s=\frac{3}{640}\cdot30^2=4{,}21875\approx4m$

jarrutusmatka vähenee

$12m-4m=8m$