Przykład 3. Oblicz:

a) $5^{-20}\cdot 5^{19}=5^{-20+19}=5^{-1}=\frac{1}{5}$

b) ${\left(7^2\right)}^{-18}\cdot {\left(7^{-9}\right)}^{-4}=7^{2\cdot \left(-18\right)}\cdot 7^{-9\cdot \left(-4\right)}=7^{-36}\cdot 7^{36}=7^{-36+36}=7^0=1$

c) ${\left(-1\frac{2}{3}\right)}^{-1}:{\left(-1\frac{2}{3}\right)}^{-3}={\left(-1\frac{2}{3}\right)}^{-1-\left(-3\right)}={\left(-1\frac{2}{3}\right)}^{-1+3}={\left(-1\frac{2}{3}\right)}^2={\left(-\frac{5}{3}\right)}^2=\frac{25}{9}=2\frac{7}{9}$

d) Dla $a\ne 0$ mamy: $\frac{a^2\cdot a^5}{{\left(a^{-2}\right)}^3}=\frac{a^{2+5}}{a^{-2\cdot 3}}=\frac{a^7}{a^{-6}}=a^{7-\left(-6\right)}=a^{13}$

e) ${\left(3\frac{1}{7}\right)}^{-12}\cdot {\left(\frac{7}{22}\right)}^{-12}={\left(3\frac{1}{7}\cdot \frac{7}{22}\right)}^{-12}={\left(\frac{22}{7}\cdot \frac{7}{22}\right)}^{-12}=1^{-12}=1$