a)

$2^4\cdot 4^3\cdot 2^{-1}=2^4\cdot {\left(2^2\right)}^3\cdot 2^{-1}=2^4\cdot 2^6\cdot 2^{-1}=2^9$

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b) 

$4^3\cdot 2^4:8^{-4}={\left(2^2\right)}^3\cdot 2^4:{\left(2^3\right)}^{-4}=2^6\cdot 2^4:2^{-12}=2^{10}:2^{-12}=2^{22}$   

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c)

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

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d)

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

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e)

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

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f)

${\left(8^{-6}:4^{-6}\right)}^{-1}={\left({\left(2^3\right)}^{-6}:{\left(2^2\right)}^{-6}\right)}^{-1}={\left(2^{-18}:2^{-12}\right)}^{-1}={\left(2^{-6}\right)}^{-1}=2^6$