a)
32^{\frac{3}{5}}=(2^5)^{\frac{3}{5}}=2^3=8
8^{\frac{2}{3}}=(2^3)^{\frac{2}{3}}=2^2=4
(\frac{1}{16})^{\frac{1}{4}}=((\frac{1}{2})^4)^{\frac{1}{4}}=\frac{1}{2}
(\frac{1}{64})^{\frac{2}{3}}=((\frac{1}{2})^6)^{\frac{2}{3}}=(\frac{1}{2})^4=\frac{1}{16}
b)
(1\frac{2}{3})^{-2}
(1\frac{4}{5})^{-2}
(1\frac{7}{8})^{-2}
c)
2^{-9}
2^{-12}
2^{-15}
d)
(32\sqrt2)^2=(2^5)^2*(2^{\frac{1}{2}})^2=2^{10}*2
(8\sqrt2)^3=(2^3)^3*\sqrt{2^3}=2^9*\sqrt{2^2*2}=2^9*2\sqrt2=2^{10}*\sqrt2
16^{\frac{5}{2}}=(2^4)^{\frac{5}{2}}=2^{10}
e)
5^{20}=(5^2)^{10}=25^{10}
2^{40}=(2^4)^{10}=16^{10}
15^{10}
f)
2^{500}=(2^5)^{100}=32^{100}
3^{300}=(3^3)^{100}=27^{100}
5^{200}=(5^2)^{100}=25^{100}