$a){\left({10}^{-1}\right)}^2={10}^{-1\cdot 2}={10}^{-2}=\frac{1}{{10}^2}=\frac{1}{100}$

$b){\left(4^{-3}\right)}^{-1}=4^{-3\cdot \left(-1\right)}=4^3=64$ $b){\left(4^{-3}\right)}^{-1}=4^{-3\cdot \left(-1\right)}=4^3=64$

$c){\left(7^{-1}\right)}^{-1}=7^{-1\cdot \left(-1\right)}=7^1=7$

$d){\left(3^{-2}\right)}^3=3^{-2\cdot 3}=3^{-6}=\frac{1}{3^6}=\frac{1}{729}$

$e){\left(2^4\right)}^{-2}=2^{4\cdot \left(-2\right)}=2^{-8}=\frac{1}{2^8}=\frac{1}{256}$

$f){\left(5^{-1}\right)}^{-2}=5^{-1\cdot \left(-2\right)}=5^2=25$