431. a) 8$$\frac{1}{2}$$: (8$$\frac{1}{6}$$⋅ 9$$\frac{1}{2}$$); б) $$\sqrt[3]{100}$$⋅ ($$\sqrt{2}$$)$$^{\frac{8}{3}}$$⋅ ($$\frac{1}{5}$$)$$^{\frac{5}{3}}$$; в) 8$$\frac{2}{3}$$:81$$^{0,75}$$; г) ($$\frac{11}{25}$$)$$^{-0,5}$$⋅ (4$$\frac{17}{27}$$)$$^{-\frac{1}{3}}$$

а) 8^1/2 : (8^1/6 ⋅ 9^1/2) = 8^1/2 : (8^1/6 ⋅ (3²)^1/2) = 8^1/2 : (8^1/6 ⋅ 3) = (2³)^1/2 : ((2³)^1/6 ⋅ 3) = 2^3/2 : (2^1/2 ⋅ 3) = 2^3/2 / (2^1/2 ⋅ 3) = 2^{3/2 - 1/2} / 3 = 2¹ / 3 = 2/3

б) ³$$\sqrt{100}$$⋅ ($$\sqrt{2}$$)$$^{\frac{8}{3}}$$⋅ ($$\frac{1}{5}$$)$$^{\frac{5}{3}}$$ = 100^1/3 ⋅ (2^1/2)^8/3 ⋅ (5^-1)^5/3 = (10²)^1/3 ⋅ 2^4/3 ⋅ 5^-5/3 = 10^2/3 ⋅ 2^4/3 ⋅ 5^-5/3 = (2⋅5)^2/3 ⋅ 2^4/3 ⋅ 5^-5/3 = 2^2/3 ⋅ 5^2/3 ⋅ 2^4/3 ⋅ 5^-5/3 = 2^{2/3 + 4/3} ⋅ 5^{2/3 - 5/3} = 2^6/3 ⋅ 5^-3/3 = 2² ⋅ 5^-1 = 4 ⋅ 1/5 = 4/5

в) 8^2/3 : 81^0,75 = (2³)^2/3 : (3⁴)^3/4 = 2² : 3³ = 4/27

г) ($$\frac{11}{25}$$)$$^{-0,5}$$⋅ (4$$\frac{17}{27}$$)$$^{-\frac{1}{3}}$$ = ($$\frac{25}{11}$$)$$^{0,5}$$ ⋅ ($$\frac{125}{27}$$)$$^{-\frac{1}{3}}$$ = $$\sqrt{\frac{25}{11}}$$ ⋅ ($$\frac{5^3}{3^3}$$)$$^{-\frac{1}{3}}$$ = $$\frac{5}{\sqrt{11}}$$ ⋅ ($$\frac{5}{3}$$)$$^{-1}$$ = $$\frac{5}{\sqrt{11}}$$ ⋅ $$\frac{3}{5}$$ = $$\frac{3}{\sqrt{11}}$$ = $$\frac{3\sqrt{11}}{11}$$