1) $$(x^{-6})^2 = x^{-6 \cdot 2} = x^{-12}$$
2) $$(-x^5)^4 = (-1)^4 \cdot (x^5)^4 = x^{5 \cdot 4} = x^{20}$$
3) $$x^4x^3 = x^{4+3} = x^7$$
4) $$((x^{-2})^2)^5 = (x^{-2 \cdot 2})^5 = (x^{-4})^5 = x^{-4 \cdot 5} = x^{-20}$$
5) $$(x^{10})^3(x^5)^4 = x^{10 \cdot 3} \cdot x^{5 \cdot 4} = x^{30} \cdot x^{20} = x^{30+20} = x^{50}$$
6) $$(-x^6)^7-(-x^3)^2:x^{15} = (-1)^7(x^6)^7 - (x^3)^2 : x^{15} = -x^{42} - x^6 : x^{15} = -x^{42} - x^{6-15} = -x^{42} - x^{-9}$$