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123456Machten

Antwoorden van de opgaven

Opgave V1

$1024$ lagen.

Opgave 1
a

$1$

b

$42 , 875$

c

$\frac{1}{81}$

d

$\frac{16}{625}$

Opgave 2
a

$81$

b

${\left(\textrm{-} 3\right)}^{4} = \textrm{-} 3 \cdot \textrm{-} 3 \cdot \textrm{-} 3 \cdot \textrm{-} 3 = 81$

c

$\textrm{-} {3}^{4} = \textrm{-} 3 \cdot 3 \cdot 3 \cdot 3 = \textrm{-} 81$

Opgave 3
a

$\frac{1}{16}$

b

$18 \frac{26}{27}$

c

$1$

d

$\textrm{-} 18$

Opgave 4
a

${3}^{95} \cdot {3}^{114} = {3}^{209}$

b

${3}^{114} / {3}^{95} = {3}^{19}$

c

${\left({3}^{12}\right)}^{5} = {3}^{60}$

d

$1$

Opgave 5
a

$4 \cdot 32 - \frac{400}{16} = 103$

b

$\frac{{\left(8 + 9\right)}^{2}}{17} - 4 = 17 - 4 = 13$

c

${\left(2 \cdot \frac{1}{8}\right)}^{3} = {\left(\frac{1}{4}\right)}^{3} = \frac{1}{64}$

Opgave 6
a

$500 - \textrm{-} 5 \cdot 5 \cdot 5 = 625$

b

$500 - \textrm{-} 5 \cdot \textrm{-} 5 \cdot \textrm{-} 5 = 625$

c

$500 + \textrm{-} 5 \cdot 5 \cdot 5 \cdot 5 = \textrm{-} 125$

d

$500 + \textrm{-} 5 \cdot \textrm{-} 5 \cdot \textrm{-} 5 \cdot \textrm{-} 5 = 1125$

Opgave 7
a

$\frac{{13}^{240}}{{\left({13}^{10}\right)}^{4} \cdot {13}^{200}} = \frac{{13}^{240}}{{13}^{40} \cdot {13}^{200}} = 1$.

b

$\frac{{4}^{99}}{{2}^{150} \cdot {\left(\textrm{-} {2}^{5}\right)}^{10}} = \frac{{\left({2}^{2}\right)}^{99}}{{2}^{150} \cdot {2}^{50}} = \frac{{2}^{198}}{{2}^{200}} = \frac{1}{{2}^{2}} = \frac{1}{4}$.

Opgave 8
a

$1 , 20 \cdot {10}^{6} / 42 \cdot {10}^{3} = 1 , 20 / 42 \cdot {10}^{3} \approx 28 , 6$ uur.

b

$12 / \left(6 , 02 \cdot {10}^{23}\right) \approx 1 , 99 \cdot {10}^{\textrm{-} 23} \approx 19 , 9 \cdot {10}^{\textrm{-} 24}$ gram.

Opgave 9
a

${4}^{5} = 1024$

b

${3}^{4} \cdot {2}^{3} = 648$

c

${\left(\frac{2}{3}\right)}^{4} = \frac{16}{81}$

d

${\left(1 \frac{3}{5}\right)}^{3} = 4 \frac{12}{125}$

e

${\left(\textrm{-} 2\right)}^{6} = 64$

f

$\textrm{-} {2}^{4} \cdot {3}^{3} = \textrm{-} 432$

Opgave 10
Opgave 11
a

${2}^{16} \cdot {\left({2}^{10}\right)}^{3} = {2}^{46}$

b

$\frac{4 \cdot {2}^{26}}{2} ^ 20 = {2}^{8}$

c

$\frac{{2}^{14} \cdot {2}^{26}}{{2}^{20}} ^ 2 = {2}^{0} = 1$

Opgave 12
a

$12 \cdot 1 , 66 \cdot {10}^{\textrm{-} 24} = 19 , 92 \cdot {10}^{\textrm{-} 24} \approx 1 , 99 \cdot {10}^{\textrm{-} 23}$ gram.

b

Uit ongeveer $\frac{12}{1 , 99 \cdot {10}^{\textrm{-} 23} \approx 6 , 02 \cdot {10}^{23}}$ atomen. (Dit getal is de constante van Avogadro.)

c

Allebei ongeveer $6 , 02 \cdot {10}^{23}$ atomen.

d

Ongeveer $\frac{{10}^{3}}{18 \cdot 1 , 66 \cdot {10}^{\textrm{-} 24}} \approx 3 , 35 \cdot {10}^{25}$.

Opgave 13
a

$\frac{1}{150 \cdot {10}^{6}} \approx 6 , 7 \cdot {10}^{\textrm{-} 9}$ AE. (Denk er om dat je antwoorden ook in de technische notatie moeten staan en dat veel decimalen of exacte waarden nu onzinnig zijn.)

b

$5 , 2 \cdot 150 \cdot {10}^{6} = 780 \cdot {10}^{6}$ km.

c

$\frac{5 , 9 \cdot {10}^{9}}{150 \cdot {10}^{6}} \approx 39$ AE.

d

Het licht legt in een jaar ongeveer $365 \cdot 24 \cdot 60 \cdot 60 \cdot 300 \cdot {10}^{6} / 1000 \approx 9 , 46 \cdot {10}^{12}$ km af. Dat is ongeveer $63072$ AE.

Opgave A1
a

${2}^{9}=512$

b

${2}^{63}$.

c

Ongeveer $9 , 22 \cdot {10}^{18}$.

d

Ongeveer $0 , 065 \cdot 9 , 22 \cdot {10}^{18} = 0 , 5993 \cdot {10}^{18} \approx 5 , 99 \cdot {10}^{17}$ gram en dat is ongeveer $5 , 99 \cdot {10}^{14} \approx 599 \cdot {10}^{12}$ kg.

e

Je moet er $9 , 22 \cdot {10}^{16}$ cm3 graan in kwijt kunnen. De hoogte wordt dan $9 , 22 \cdot {10}^{16} / 25 = 3 , 69 \cdot {10}^{15}$ cm en dat is $3 , 69 \cdot {10}^{10}$ km oftwel $36,9$ miljard km.

Opgave A2
a

$1+2+{2}^{2}+{2}^{3}+{2}^{4}+{2}^{5}+{2}^{6}+{2}^{7}+{2}^{8}+{2}^{9}=1023$

b

Daar komt ook $1023$ uit.

c

Stel $s=1+2+{2}^{2}+{2}^{3}+...+{2}^{62}+{2}^{63}$, dan is $2s=2+{2}^{2}+{2}^{3}+...+{2}^{62}+{2}^{63}+{2}^{64}$. Daaruit volgt $s=2s-s={2}^{64}-1$.

Opgave T1
a

${2}^{7} = 128$

b

${\left(\textrm{-} 2\right)}^{7} = \textrm{-} 128$

c

$\textrm{-} {2}^{7} = \textrm{-} 128$

d

${\left(\frac{2}{3}\right)}^{4} = \frac{16}{81}$

e

${3}^{6} = 729$

Opgave T2
a

$\textrm{-} 4 \cdot {2}^{3} - {\left(\frac{3}{4}\right)}^{3} = \textrm{-} 32 \frac{27}{64}$

b

$\textrm{-} 2 \cdot {\left(\textrm{-} 4\right)}^{4} = \textrm{-} 512$

c

${\left(\frac{2}{4}\right)}^{3} \cdot 3 + {2}^{6} = 64 \frac{3}{8}$

d

$\textrm{-} {4}^{4} + {3}^{5} \cdot {2}^{2} = 716$

Opgave T3
a

${\left({9}^{20}\right)}^{4} / \left({3}^{2} \cdot {9}^{16}\right) = {9}^{63}$

b

${6}^{16} \cdot {6}^{12} = {6}^{28}$

c

${\left({4}^{17}\right)}^{4} = {4}^{68}$

d

${12}^{14} \cdot {12}^{80} / {12}^{56} = {12}^{38}$