Van $3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$ .

$3^{2} \cdot 3^{6}$	$=$	$3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$
	$=$	$3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3$
	$=$	$3^{8}$

$3^{5} \cdot 3^{7} = 3^{5 + 7} = 3^{12}$
$3^{4} \cdot 3^{8} = 3^{4 + 8} = 3^{12}$
$3^{5} \cdot 3^{6} = 3^{5 + 6} = 3^{11}$
$3^{4} \cdot 3 = 3^{4 + 1} = 3^{5}$

$3 \cdot 3^{5} \cdot 3^{7} = 3^{1 + 5 + 7} = 3^{13}$
$3^{3} \cdot 3^{4} \cdot 3^{8} = 3^{3 + 4 + 8} = 3^{15}$
$3^{4} \cdot 3^{5} \cdot 3^{6} = 3^{4 + 5 + 6} = 3^{15}$
$3^{4} \cdot 3 \cdot 3^{3} = 3^{4 + 1 + 3} = 3^{8}$

Nee, $3^{5} = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 243$ en $5^{3} = 5 \cdot 5 \cdot 5 = 125$ .

$2^{7} = 128$ , $7^{2} = 49$ , $4^{5} = 1024$ , $5^{4} = 625$

Het kwadraat van 7.

$6^{4} = 1296$

$\begin{array}{l} 0^{3} = 0; 1^{3} = 1; 2^{3} = 8; 3^{3} = 27 \\ 4^{3} = 64; 5^{3} = 125; 6^{3} = 216 \end{array}$

$\begin{array}{l} 0^{4} = 0; 1^{4} = 1; 2^{4} = 16; 3^{4} = 81 \\ 4^{4} = 256 \end{array}$

$0^{5} = 0; 1^{5} = 1; 2^{5} = 32; 3^{5} = 243$

$64 = 2^{6}$ , $64 = 4^{3}$ , $64 = 8^{2}$

$1000 = 10^{3}$ , $8 = 2^{3}$ , $8000 = 20^{3}$

$\frac{2}{5} \cdot \frac{2}{5} \cdot \frac{2}{5} = \frac{8}{125}$	$\frac{1}{10} \cdot \frac{1}{10} \cdot \frac{1}{10} \cdot \frac{1}{10} = \frac{1}{10.000}$
$\frac{3}{4} \cdot \frac{3}{4} \cdot \frac{3}{4} = \frac{27}{64}$	$\frac{11}{10} \cdot \frac{11}{10} \cdot \frac{11}{10} = \frac{1331}{1000}$
$\frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} = \frac{16}{81}$	$\frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} \cdot \frac{1}{5} = \frac{1}{625}$

$- 3 \cdot - 3 \cdot - 3 \cdot - 3 = 81$

$- 2 \cdot -2 \cdot -2 = - 8$

De hoofdeigenschap

$2^{2} \cdot 2^{6} = 2^{2 + 6} = 2^{8}$	$2^{5} \cdot 4 = 2^{5} \cdot 2^{2} = 2^{5 + 2} = 2^{7}$
$3^{2} \cdot 3^{6} = 3^{2 + 6} = 3^{8}$	$3^{2} \cdot 27 = 3^{2} \cdot 3^{3} = 3^{2 + 3} = 3^{5}$
$5^{2} \cdot 5^{3} = 5^{2 + 3} = 5^{5}$	$5^{2} \cdot 25 = 5^{2} \cdot 5^{2} = 5^{2 + 2} = 5^{4}$

$10^{3} \cdot 10^{3} = 10^{3 + 3} = 10^{6}$	$10^{11} \cdot 10.000 = 10^{11} \cdot 10^{4} = 10^{11 + 4} = 10^{15}$
$10^{n + 3}$	$10^{n} \cdot 100 = 10^{n} \cdot 10^{2} = 10^{n + 2}$
$10^{n + m}$	$10^{n + 1}$

${(\frac{1}{2})}^{3} \cdot {(\frac{1}{2})}^{3} = {(\frac{1}{2})}^{3 + 3} = {(\frac{1}{2})}^{6}$

${(\frac{3}{4})}^{3} \cdot {(\frac{3}{4})}^{7} = {(\frac{3}{4})}^{3 + 7} = {(\frac{3}{4})}^{10}$

Macht van een macht

${(2^{3})}^{3} = 2^{3} \cdot 2^{3} \cdot 2^{3} = 2^{3 \cdot 3} = 2^{9}$
${(2^{3})}^{4} = 2^{3} \cdot 2^{3} \cdot 2^{3} \cdot 2^{3} = 2^{4 \cdot 3} = 2^{12}$ ${(2^{4})}^{0} = 2^{4 \cdot 0} = 2^{0}$

${(3^{2})}^{3} = 3^{2} \cdot 3^{2} \cdot 3^{2} = 3^{3 \cdot 2} = 3^{6}$
${(3^{3})}^{4} = 3^{3} \cdot 3^{3} \cdot 3^{3} \cdot 3^{3} = 3^{4 \cdot 3} = 3^{12}$
${(3^{4})}^{0} = 3^{4 \cdot 0} = 3^{0}$

${(2^{4})}^{10} = 2^{4 \cdot 10} = 2^{40}$	${(2^{5})}^{10} = 2^{5 \cdot 10} = 2^{50}$
${(3^{2})}^{10} = 3^{2 \cdot 10} = 3^{20}$	${(3^{3})}^{15} = 3^{3 \cdot 15} = 3^{45}$
${(2^{2})}^{11} = 2^{2 \cdot 11} = 2^{22}$	${(2^{3})}^{15} = 2^{3 \cdot 15} = 2^{45}$
$a^{p \cdot 3} = a^{3 p}$	$a^{p \cdot q} = a^{p q}$

$64 = 2^{6}$ ; $64 = 4^{3}$ ; $64 = 8^{2}$

${(2^{6})}^{3} = 2^{6 \cdot 3} = 2^{18}$ ; ${(4^{3})}^{3} = 4^{3 \cdot 3} = 4^{9}$ ; ${(8^{2})}^{3} = 8^{2 \cdot 3} = 8^{6}$

$f^{8} = {(f^{4})}^{2} = 25^{2} = 625$
$f^{2} = \sqrt{f^{4}} = \sqrt{25} = 5$
$f^{6} = {(f^{2})}^{3} = 5^{3} = 125$

Machten delen

$3 \cdot 12 = 36$ , dus $36 : 3 = 12$

$3^{6}$ ; $3^{6}$
$3^{8}$ ; $3^{8}$
$3^{10}$ ; $3^{10}$
$3^{20}$ ; $3^{20}$

$3^{40} : 3^{10} = 3^{40 - 10} = 3^{30}$	$3^{40} : 3^{5} = 3^{40 - 5} = 3^{35}$
$3^{40} : 3^{40} = 3^{40 - 40} = 3^{0}$	$3^{40} : 3^{30} = 3^{40 - 30} = 3^{10}$

${(\frac{1}{2})}^{40} : {(\frac{1}{2})}^{10} = {(\frac{1}{2})}^{40 - 10} = {(\frac{1}{2})}^{30}$
${(\frac{1}{2})}^{40} : {(\frac{1}{2})}^{30} = {(\frac{1}{2})}^{40 - 30} = {(\frac{1}{2})}^{10}$
${(\frac{1}{2})}^{40} : {(\frac{1}{2})}^{15} = {(\frac{1}{2})}^{40 - 15} = {(\frac{1}{2})}^{25}$
${(\frac{1}{2})}^{40} : {(\frac{1}{2})}^{40} = {(\frac{1}{2})}^{40 - 40} = {(\frac{1}{2})}^{0}$

Machten van breuken

$\frac{2^{4}}{3^{4}} = \frac{16}{81}$	$\frac{3^{5}}{2^{5}} = \frac{243}{32}$
$\frac{2^{6}}{3^{6}} = \frac{64}{729}$	$\frac{3^{7}}{2^{7}} = \frac{2187}{128}$

$\frac{3^{6}}{10^{6}} = \frac{729}{1.000.000}$	$\frac{2^{7}}{10^{7}} = \frac{128}{10.000.000}$
$\frac{1^{7}}{3^{7}} = \frac{1}{2187}$	$\frac{10^{6}}{3^{6}} = \frac{1.000.000}{729}$

Machten met dezelfde exponent

100

1000

10.000

$2^{3} \cdot 5^{3} = 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5 \cdot 5 = 2 \cdot 5 \cdot 2 \cdot 5 \cdot 2 \cdot 5 =$ $10 \cdot 10 \cdot 10 = 1000$
${0,2}^{3} \cdot 10^{3} = 0,2 \cdot 0,2 \cdot 0,2 \cdot 10 \cdot 10 \cdot 10 = 0,2 \cdot 10 \cdot 0,2 \cdot 10 \cdot 0,2 \cdot 10 = 2 \cdot 2 \cdot 2 = 2^{3}$

$2^{5} \cdot 3^{5} = {(2 \cdot 3)}^{5} = 6^{5}$	$20^{6} \cdot {0,1}^{6} = {(20 \cdot 0,1)}^{6} = 2^{6}$
${(\frac{1}{9})}^{5} \cdot 3^{5} = {(\frac{1}{9} \cdot 3)}^{5} = {(\frac{1}{3})}^{5}$	$20^{7} \cdot {(\frac{1}{4})}^{7} = {(20 \cdot \frac{1}{4})}^{7} = 5^{7}$

Linkerkolom:
${(\frac{1}{9})}^{3} \cdot 3^{3} = {(\frac{1}{9} \cdot 3)}^{3} = {(\frac{1}{3})}^{3} = \frac{1}{27}$
${(\frac{1}{9})}^{3} \cdot 3^{5} = {(\frac{1}{9})}^{3} \cdot 3^{3} \cdot 3^{2} = \frac{1}{27} \cdot 9 = \frac{1}{3}$
${(\frac{1}{2})}^{10} \cdot 2^{10} = {(\frac{1}{2} \cdot 2)}^{10} = 1^{10} = 1$

Rechterkolom:
${(\frac{1}{9})}^{3} \cdot 3^{4} = {(\frac{1}{9})}^{3} \cdot 3^{3} \cdot 3 = \frac{1}{27} \cdot 3 = \frac{1}{9}$
${(\frac{1}{9})}^{3} \cdot 3^{6} = {(\frac{1}{9})}^{3} \cdot 3^{3} \cdot 3^{3} = \frac{1}{27} \cdot 27 = 1$
${(\frac{1}{4})}^{4} \cdot {(\frac{2}{3})}^{4} = {(\frac{1}{4} \cdot \frac{2}{3})}^{4} = {(\frac{1}{6})}^{4} = \frac{1}{1296}$

12s

13s

Linkerkolom:
${(2 \cdot 5)}^{5} = 10^{5} = 100.000$
${(\frac{1}{9} \cdot 9)}^{30} = 1^{30} = 1$
${(\frac{1}{2} \cdot 2)}^{10} = 1^{10} = 1$
${(\frac{1}{2})}^{5} \cdot 2^{5} \cdot 2^{5} = 1^{5} \cdot 2^{5} = 1 \cdot 2^{5} = 32$

Rechterkolom:
$2^{5} \cdot 5^{5} \cdot 5 = 10^{5} \cdot 5 = 500.000$
${(\frac{1}{9})}^{30} \cdot 9^{30} \cdot 9 = 1^{30} = 1^{30} \cdot 9 = 9$
${(\frac{1}{2})}^{10} \cdot 20^{10} \cdot 10^{10} = 1 \cdot 10^{10} = 10.000.000.000$
${(\frac{1}{2})}^{5} \cdot {(\frac{1}{2})}^{5} \cdot 2^{5} = {(\frac{1}{2})}^{5} \cdot 1^{5} = {(\frac{1}{2})}^{5} = \frac{1^{5}}{1^{5}} = \frac{1}{32}$

Oefenen: alles door elkaar

${(\frac{1}{10})}^{3} = \frac{1^{3}}{10^{3}} = \frac{1}{1000}$	$1.000.000$	$\frac{4^{3}}{5^{3}} = \frac{64}{125}$
$81 \cdot 1.000.000 = 81.000.000$	$1$	$1$
$1 \cdot 0 = 0$	$6 \cdot 1 = 6$	${(3 \cdot \frac{1}{2})}^{30} = {(1 \frac{1}{2})}^{30}$

${(2 \cdot 3)}^{3} = 6^{3}$	$2^{4 + 3} = 2^{7}$
$2^{4 - 3} = 2^{1}$	$2^{40 - 30} = 2^{10}$
$2^{40 - 40} = 2^{0}$	${(2 \cdot 3)}^{10} = 6^{10}$

14s

15s

4096 is het kwadraat van $2^{6} = 64$
4096 is de derde macht van $2^{4} = 16$
4096 is de vierde macht van $2^{3} = 8$
4096 is geen vijfde macht (van een geheel getal)

$4096 \cdot 4 = 2^{12} \cdot 2^{2} = 2^{12 + 2} = 2^{14} = {(2^{2})}^{7} = 4^{7}$
$4096 \cdot {(\frac{1}{2})}^{6} = 2^{12} \cdot {(\frac{1}{2})}^{6} = 2^{6} \cdot 2^{6} \cdot {(\frac{1}{2})}^{6} = 2^{6} \cdot 1^{6} = 2^{6} = {(2^{2})}^{3} = 4^{3}$
$4096 \cdot \frac{1}{4} = 2^{12} : 2^{2} = 2^{12 - 2} = 2^{10} = {(2^{2})}^{5} = 4^{5}$
$4096 \cdot {(\frac{1}{4})}^{6} = 2^{12} \cdot {(\frac{1}{4})}^{6} = 2^{12} : {(2^{2})}^{6} = 2^{12} : 2^{12} = 2^{0} = 4^{0}$

$4^{6 - 7 + 3} = 4^{2}$	${(3^{4})}^{6} = 3^{24}$	${(3^{3})}^{5} \cdot {(3^{2})}^{4} = 3^{15} \cdot 3^{8} = 3^{23}$
$2^{20} : {(2^{2})}^{5} \cdot {(2^{3})}^{2} = 2^{20 - 10 + 6} = 2^{16}$	$2^{12}$	${(10^{2})}^{p} \cdot 10^{4} = 10^{2 p + 4}$
$‐ 10.000$	${(10^{3})}^{p} : 10^{a} = 10^{3 p - a}$

$10^{4}$	$10^{5}$
$10^{6}$	$10^{7}$
$10^{4}$	$10^{5}$

$2^{27} = 67.108.864 \cdot 2 = 134.217.728$
$2^{25} = 67.108.864 : 2 = 33.554.432$