2.
[tex]a) \ 3^{13} : 3^{2}:3^{3} = 3^{13-2}:3^{3} = 3^{11}:3^{3} = 3^{11-3} = 3^{8}\\\\\\b) \ \frac{8^{9}\cdot8\cdot8^{0}}{8^{8}} = \frac{8^{9+1+0}}{8^{8}} = \frac{8^{10}}{8^{8}} = 8^{10-8} = 8^{2}\\\\\\c) \ \frac{(a^{4})^{4}\cdot a^{0}\cdot a^{16}}{a^{8}:a^{6}}=\frac{a^{4\cdot4}\cdot1\cdot a^{12}}{a^{8-6}} = \frac{a^{16+16}}{a^{2}} = a^{32-2} = a^{30}, \ \ \ (a\neq 0)[/tex]
[tex]d) \ \frac{(x^{3})^{5}\cdot(x^{15})^{0}}{((x^{5})^{2}:x^{10})^{4}} =\frac{x^{3\cdot5}\cdot1}{(x^{5\cdot2}:x^{10})^{4}}=\frac{x^{15}}{(x^{10}:x^{10})^{4}} = \frac{x^{15}}{1^{4}} = \frac{x^{15}}{1} = x^{15}, \ \ \ (x\neq 0)[/tex]
[tex]e) \ [179^{6}\cdot(179^{5}:179^{2})]:179^{4} = (179^{6}\cdot179^{5-2}):179^{4} = (179^{6}\cdot179^{3}):179^{4} =\\\\=179^{6+3-4} = 179^{5}[/tex]
3.
[tex]a) \ 16\cdot2^{5}\cdot2^{7} = 2^{4}\cdot2^{5}\cdot2^{7} = 2^{4+5+7} = 2^{16}\\\\b) \ (64\cdot32)^{5} = (2^{6}\cdot2^{5})^{5} = (2^{6+5})^{5} =(2^{11})^{5} = 2^{11\cdot5} = 2^{55}[/tex]
4.
[tex]a) \ 81^{4}\cdot21^{3}:7^{3}:3^{6} = (3^{4})^{4}\cdot(21:7)^{3}:3^{6} = 3^{4\cdot4}\cdot3^{3}:3^{6} =3^{16+3-6} = 3^{13}\\\\b) \ \frac{9^{7}\cdot27^{5}}{81^{2}} = \frac{(3^{2})^{7}\cdot(3^{3})^{5}}{(3^{4})^{2}} = \frac{3^{2\cdot7}\cdot3^{3\cdot5}}{3^{4\cdot2}}=\frac{3^{14}\cdot3^{15}}{3^{8}} = 3^{14+15-8} = 3^{21}[/tex]
Wyjaśnienie
Skorzystano z praw potęgowania:
[tex]a^{0} = 1 \ \ \ dla \ a\neq 0\\\\a^{1} = a \ \ \ dla \ kazdego \ a\\\\a^{m}\cdot a^{n} = a^{m+n}\\\\a^{m}:a^{n} = a^{m-n}\\\\a^{n}:b^{n} = (a:b)^{n}\\\\(a^{n})^{m} = a^{n\cdot m}[/tex]
