[tex]a) \ 3^{-2}\cdot(3^{\frac{2}{3}})^{3}\cdot27=3^{-2}\cdot3^{\frac{2}{\not3}\cdot\not3}\cdot3^{3}=3^{-2}\cdot3^{2}\cdot3^{3}=3^{-2+2+3}=3^{3}=\boxed{27}[/tex]
[tex]b) \ \dfrac{32^{\frac{1}{2}}\cdot\sqrt{8}}{8^{-2}}=\dfrac{(2^{5})^{\frac{1}{2}}\cdot8^{\frac{1}{2}}}{(2^{3})^{-2}}=\dfrac{2^{\frac{5}{2}}\cdot(2^{3})^{\frac{1}{2}}}{2^{-6}}=\dfrac{2^{\frac{5}{2}}\cdot2^{\frac{3}{2}}}{2^{-6}}=\\\\=\dfrac{2^{\frac{8}{2}}}{2^{-6}}=2^{4}:2^{-6}=2^{4-(-6)}=2^{10}=\boxed{1024}[/tex]
[tex]c) \ \dfrac{81^{\frac{1}{3}}\cdot27^{\frac{1}{3}}}{\sqrt{27}}=\dfrac{(3^{4})^{\frac{1}{3}}\cdot(3^{3})^{\frac{1}{3}}}{27^{\frac{1}{2}}}=\dfrac{3^{\frac{4}{3}}\cdot3^{\frac{3}{3}}}{(3^{3})^{\frac{1}{2}}}=\dfrac{3^{\frac{7}{3}}}{3^{\frac{3}{2}}}=\\\\=\dfrac{3^{\frac{14}{6}}}{3^{\frac{9}{6}}}=3^{\frac{14}{6}-\frac{9}{6}}=3^{\frac{5}{6}}=\sqrt[6]{3^{5}}=\boxed{\sqrt[6]{243}}[/tex]
[tex]d) \ 2^{\frac{5}{3}}\cdot\sqrt[3]{2^{4}}=2^{\frac{5}{3}}\cdot2^{\frac{4}{3}}=2^{\frac{5}{3}+\frac{4}{3}}=2^{\frac{9}{3}}=2^{3}=\boxed{8}[/tex]
Skorzystano ze wzorów:
[tex]a^{m}\cdot a^{n}=a^{m+n}[/tex]
[tex]a^{m}:a^{n}=a^{m-n}, \ a\neq 0[/tex]
[tex](a^{m})^{n}=a^{m\cdot n}[/tex]
[tex]a^{\frac{1}{n}}=\sqrt[n]{a}, \ a\geq 0, \ n\in\mathbb{N}[/tex]
[tex]a^{\frac{k}{n}}=\sqrt[n]{a^{k}}, \ a\geq 0,\ n,k\in\mathbb{N}[/tex]
