Решение примеров

1. $$x^{-5} \cdot x = x^{-5+1} = x^{-4}$$
2. $$x^{-6} : x = x^{-6-1} = x^{-7} = \frac{1}{x^7}$$
3. $$(x^{-5})^{-1} = x^{-5 \cdot (-1)} = x^{5}$$
4. $$\frac{x^{-4}}{x^{-3}} = x^{-4 - (-3)} = x^{-4 + 3} = x^{-1} = \frac{1}{x}$$
5. $$x^{5} \cdot x^{-2} \cdot x = x^{5 + (-2) + 1} = x^{5 - 2 + 1} = x^{4}$$
6. $$(x^{-2})^4 \cdot x^3 = x^{-2 \cdot 4} \cdot x^3 = x^{-8} \cdot x^3 = x^{-8 + 3} = x^{-5} = \frac{1}{x^5}$$
7. $$\frac{x^{-5} \cdot x^8}{x^4 \cdot x^{-7}} = \frac{x^{-5+8}}{x^{4+(-7)}} = \frac{x^3}{x^{-3}} = x^{3-(-3)} = x^{3+3} = x^6$$
8. $$(5x^4)^{-2} = 5^{-2} \cdot (x^4)^{-2} = \frac{1}{5^2} \cdot x^{4 \cdot (-2)} = \frac{1}{25} \cdot x^{-8} = \frac{1}{25x^8}$$
9. $$(0.5m^3n^{-5})^2 = (0.5)^2 \cdot (m^3)^2 \cdot (n^{-5})^2 = 0.25 \cdot m^{3 \cdot 2} \cdot n^{-5 \cdot 2} = 0.25m^6n^{-10} = \frac{0.25m^6}{n^{10}} = \frac{m^6}{4n^{10}}$$
10. $$\left(-\frac{c}{2}\right)^{-2} = \left(-\frac{2}{c}\right)^{2} = \frac{(-2)^2}{c^2} = \frac{4}{c^2}$$
11. $$\left(-\frac{2mn^3}{k^5}\right)^{-5} = \left(-\frac{k^5}{2mn^3}\right)^{5} = -\frac{k^{5\cdot 5}}{2^5m^5n^{3 \cdot 5}} = -\frac{k^{25}}{32m^5n^{15}}$$
12. $$6^{-4} \cdot 36 = 6^{-4} \cdot 6^2 = 6^{-4+2} = 6^{-2} = \frac{1}{6^2} = \frac{1}{36}$$
13. $$25^{-4} : 5^{-6} = (5^2)^{-4} : 5^{-6} = 5^{-8} : 5^{-6} = 5^{-8 - (-6)} = 5^{-8 + 6} = 5^{-2} = \frac{1}{5^2} = \frac{1}{25}$$
14. $$9^{-4} \cdot 3^{4} = (3^2)^{-4} \cdot 3^4 = 3^{-8} \cdot 3^4 = 3^{-8 + 4} = 3^{-4} = \frac{1}{3^4} = \frac{1}{81}$$
15. $$\frac{125 \cdot 25^{-5}}{5^{-6}} = \frac{5^3 \cdot (5^2)^{-5}}{5^{-6}} = \frac{5^3 \cdot 5^{-10}}{5^{-6}} = \frac{5^{3-10}}{5^{-6}} = \frac{5^{-7}}{5^{-6}} = 5^{-7 - (-6)} = 5^{-7 + 6} = 5^{-1} = \frac{1}{5}$$