Szczegółowe wyjaśnienie:
[tex]a)\ 4^5\cdot5^6=4^5\cdot5^{5+1}=4^5\cdot5^5\cdot5^1=(4\cdot5)^5\cdot5=20^5\cdot5=3200000\cdot5=16\ 000\ 000[/tex]
[tex]b)\ \left(\dfrac{3}{5}\right)^5\cdot\left(\dfrac{5}{3}\right)^6=\left(\dfrac{3}{5}\right)^5\cdot\left(\dfrac{5}{3}\right)^5\cdot\dfrac{5}{3}=\left(\dfrac{3\!\!\!\!\diagup}{5\!\!\!\!\diagup}\cdot\dfrac{5\!\!\!\!\diagup}{3\!\!\!\!\diagup}\right)^5\cdot\dfrac{5}{3}=1^5\cdot\dfrac{5}{3}=1\cdot\dfrac{5}{3}=\dfrac{5}{3}[/tex]
[tex]c)\ \left(\dfrac{4}{9}\right)^7:\left(\dfrac{2}{9}\right)^8=\left(\dfrac{4}{9}\right)^7\cdot\left(\dfrac{9}{2}\right)^8=\left(\dfrac{4}{9}\right)^7\cdot\left(\dfrac{9}{2}\right)^7\cdot\dfrac{9}{2}=\left(\dfrac{4\!\!\!\!\diagup^2}{9\!\!\!\!\diagup}\cdot\dfrac{9\!\!\!\!\diagup}{2\!\!\!\!\diagup_1}\right)^7\cdot\dfrac{9}{2}\\\\=2^7\cdot\dfrac{9}{2}=2^6\cdot2\!\!\!\!\diagup\cdot\dfrac{9}{2\!\!\!\!\diagup}=64\cdot9=576[/tex]
[tex]d)\ \left(1\dfrac{4}{9}\right)^5:\left(4\dfrac{1}{3}\right)^4=\left(\dfrac{13}{9}\right)^5:\left(\dfrac{13}{3}\right)^4=\dfrac{13}{9}\cdot\left(\dfrac{13}{9}\right)^4\cdot\left(\dfrac{3}{13}\right)^4\\\\=\dfrac{13}{9}\cdot\left(\dfrac{13\!\!\!\!\!\diagup}{9\!\!\!\!\diagup_3}\cdot\dfrac{3\!\!\!\!\diagup^1}{13\!\!\!\!\!\diagup}\right)^4=\dfrac{13}{9}\cdot\left(\dfrac{1}{3}\right)^4=\dfrac{13}{9}\cdot\dfrac{1}{81}=\dfrac{13}{729}[/tex]
Korzystamy z twierdzenia:
[tex]a^n\cdot a^m=a^{n+m}[/tex]
