$a){\left(-2\right)}^5=-32$

${\left(-2\right)}^{-5}=\frac{1}{{\left(-2\right)}^5}=-\frac{1}{32}$

$2^{-5}=\frac{1}{2^5}=\frac{1}{32}$

$b){\left(\frac{1}{3}\right)}^{-2}=3^2=9$

${\left(-\frac{1}{3}\right)}^{-2}={\left(-3\right)}^2=9$

${\left(-\frac{1}{3}\right)}^{-3}={\left(-3\right)}^{-3}=-27$

 

$c){\left(\sqrt{3}\right)}^4=3^2=9$

${\left(\sqrt{3}\right)}^{-2}=\frac{1}{{\left(\sqrt{3}\right)}^2}=\frac{1}{3}$

${\left(\sqrt{3}\right)}^{-6}=\frac{1}{{\left(\sqrt{3}\right)}^6}=\frac{1}{3^3}=\frac{1}{27}$

 

$d){\left(\sqrt{2}\right)}^6=2^3=8$

${\left(\sqrt{2}\right)}^7={\left(\sqrt{2}\right)}^6\cdot \sqrt{2}=2^3\cdot \sqrt{2}=8\sqrt{2}$

${\left(\sqrt{2}\right)}^{-8}=\frac{1}{{\left(\sqrt{2}\right)}^8}=\frac{1}{2^4}=\frac{1}{16}$