1. 1) $$x^{-10} \cdot x^7 = x^{-10+7} = x^{-3}$$
2) $$x^{-5} : x^{-12} = x^{-5 - (-12)} = x^{-5+12} = x^7$$
3) $$(x^4y^6z^{-5})^{-9} = x^{4 \cdot (-9)} y^{6 \cdot (-9)} z^{-5 \cdot (-9)} = x^{-36}y^{-54}z^{45}$$
4) $$\left(\frac{x^{-4}}{y^5}\right)^{-4} \cdot \left(\frac{x^{-8}}{y^{-5}}\right)^{-10} = \frac{x^{(-4) \cdot (-4)}}{y^{5 \cdot (-4)}} \cdot \frac{x^{(-8) \cdot (-10)}}{y^{(-5) \cdot (-10)}} = \frac{x^{16}}{y^{-20}} \cdot \frac{x^{80}}{y^{50}} = x^{16+80} y^{20-50} = x^{96}y^{-30}$$