$a)2\cdot 2^4\cdot 2^5=2^{1+4+5}=2^{10}$

$b)2^{-4}:2^{-6}=2^{-4-\left(-6\right)}=2^{-4+6}=2^2$

$c){\left(2^2\right)}^3\cdot 2=2^{2\cdot 3}\cdot 2=2^6\cdot 2=2^{6+1}=2^7$

$d)2\cdot 2^{15}:2^3=2^{1+15}:2^3=2^{16}:2^3=2^{16-3}=2^{13}$

$e)2^{-6}:\left(2^6:2^{-4}\right)=2^{-6}:2^{6-\left(-4\right)}=2^{-6}:2^{6+4}=2^{-6}:2^{10}=2^{-6-10}=2^{-16}$

$f)4^3\cdot 2^4:8^{-4}={\left(2^2\right)}^3\cdot 2^4:{\left(2^3\right)}^{-4}=2^6\cdot 2^4:2^{-12}=2^{6+4-\left(-12\right)}=2^{22}$  

$g)\left({16}^{-2}:2^{-3}\right)\cdot 4=\left({\left(2^4\right)}^{-2}:2^{-3}\right)\cdot 2^2=\left(2^{4\cdot \left(-2\right)}:2^{-3}\right)\cdot 2^2=2^{-8-\left(-3\right)}\cdot 2^2=2^{-8+3}\cdot 2^2=2^{-5}\cdot 2^2=2^{-5+2}=2^{-3}$

$h){\left(4^3\right)}^{-2}:{64}^{-3}={\left({\left(2^2\right)}^3\right)}^{-2}:{\left(2^6\right)}^{-3}=2^{2\cdot 3\cdot \left(-2\right)}:2^{6\cdot \left(-3\right)}=2^{-12}:2^{-18}=2^{-12-\left(-18\right)}=2^{-12+18}=2^6$

$i){32}^{-3}\cdot {\left(2^{-3}\right)}^{-5}={\left(2^5\right)}^{-3}\cdot 2^{\left(-3\right)\cdot \left(-5\right)}=2^{5\cdot \left(-3\right)}\cdot 2^{15}=2^{-15}\cdot 2^{15}=2^{-15+15}=2^0$

$j){\left(8^{-6}:4^{-6}\right)}^{-1}={\left({\left(8:4\right)}^{-6}\right)}^{-1}={\left(2^{-6}\right)}^{-1}=2^{\left(-6\right)\cdot \left(-1\right)}=2^6$