[tex]a)\ \ \dfrac{125\cdot15}{3^3\cdot5^4}=\dfrac{5^3\cdot(3\cdot5)^1}{3^3\cdot5^4}= \dfrac{5^3\cdot3^1\cdot5^1}{3^3\cdot5^4}=\dfrac{5^{3+1}\cdot3^1}{3^3\cdot5^4}=\dfrac{\not5^4\cdot3^1}{3^3\cdot\not5^4}=\dfrac{3^1}{3^3}=\\\\=3^{1-3}=3^{-2}=(\frac{1}{3})^2=\frac{1}{9}[/tex]
[tex]b)\ \ \dfrac{(3^5)^4\cdot6^4}{9^7\cdot4^2}=\dfrac{3^2^0\cdot6^4}{(3\cdot3)^7\cdot(2^2)^2}=\dfrac{3^2^0\cdot(3\cdot2)^4}{3^7\cdot3^7\cdot2^4}=\dfrac{3^2^0\cdot3^4\cdot\not2^4}{3^{7+7}\cdot\not2^4}=\dfrac{3^2^0\cdot3^4}{3^1^4}=\\\\\\=\dfrac{3^{20+4}}{3^1^4}=\dfrac{3^2^4}{3^1^4}=3^{24-14}=3^{10}[/tex]
[tex]c)\ \ \dfrac{9^3\cdot4^4}{6^1^0}=\dfrac{9^3\cdot4^3\cdot4}{6^1^0}=\dfrac{(9\cdot4)^3\cdot4}{6^1^0}=\dfrac{36^3\cdot4}{6^1^0}=\dfrac{(6^2)^3\cdot4}{6^1^0}=\dfrac{6^6\cdot4}{6^1^0}=\dfrac{4}{6^{10-6}}=\\\\\\=\dfrac{4}{6^4}=\dfrac{4}{1296}=\dfrac{1}{324}[/tex]
[tex]d)\ \ \dfrac{0,25^3:0,5^3}{5^3}=\dfrac{(0,25:0,5)^3}{5^3}=\dfrac{0,5^3}{5^3}=(0,5:5)^3=0,1^3=0,001\\\\\\e)\ \ \dfrac{(8^5\cdot4^3)^2\cdot25^2}{10^4\cdot2^1^0}=\dfrac{(8^5)^2\cdot(4^3)^2\cdot(5^2)^2}{(5\cdot2)^4\cdot2^1^0}=\dfrac{8^1^0\cdot4^6\cdot\not5^4}{\not5^4\cdot2^4\cdot2^1^0}=\dfrac{(2^3)^1^0\cdot(2^2)^6}{2^{4+10}}=\\\\\\=\dfrac{2^3^0\cdot2^1^2}{2^1^4}=\dfrac{2^{30+12}}{2^1^4}=\dfrac{2^4^2}{2^1^4}=2^{42-14}=2^{28}[/tex]
[tex]f)\ \ \dfrac{64^2\cdot36^4}{6^3\cdot2^6}=\dfrac{(2^6)^2\cdot(6^2)^4}{6^3\cdot2^6}=\dfrac{2^1^2\cdot6^8}{6^3\cdot2^6}=2^{12-6}\cdot6^{8-3}=2^6\cdot6^5=2\cdot2^5\cdot6^5 =\\\\\\=2\cdot(2\cdot6)^5=2\cdot12^5\\\\\\Zastosowane\ \ wzory\\\\a^m\cdot a^n=a^{m+n}\\\\a^m:a^n=a^{m-n}\\\\a^n\cdot b^n=(a\cdot b)^n\\\\(a^m)^n=a^{m\cdot n}[/tex]
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[tex]a)\ \ \dfrac{125\cdot15}{3^3\cdot5^4}=\dfrac{5^3\cdot(3\cdot5)^1}{3^3\cdot5^4}= \dfrac{5^3\cdot3^1\cdot5^1}{3^3\cdot5^4}=\dfrac{5^{3+1}\cdot3^1}{3^3\cdot5^4}=\dfrac{\not5^4\cdot3^1}{3^3\cdot\not5^4}=\dfrac{3^1}{3^3}=\\\\=3^{1-3}=3^{-2}=(\frac{1}{3})^2=\frac{1}{9}[/tex]
[tex]b)\ \ \dfrac{(3^5)^4\cdot6^4}{9^7\cdot4^2}=\dfrac{3^2^0\cdot6^4}{(3\cdot3)^7\cdot(2^2)^2}=\dfrac{3^2^0\cdot(3\cdot2)^4}{3^7\cdot3^7\cdot2^4}=\dfrac{3^2^0\cdot3^4\cdot\not2^4}{3^{7+7}\cdot\not2^4}=\dfrac{3^2^0\cdot3^4}{3^1^4}=\\\\\\=\dfrac{3^{20+4}}{3^1^4}=\dfrac{3^2^4}{3^1^4}=3^{24-14}=3^{10}[/tex]
[tex]c)\ \ \dfrac{9^3\cdot4^4}{6^1^0}=\dfrac{9^3\cdot4^3\cdot4}{6^1^0}=\dfrac{(9\cdot4)^3\cdot4}{6^1^0}=\dfrac{36^3\cdot4}{6^1^0}=\dfrac{(6^2)^3\cdot4}{6^1^0}=\dfrac{6^6\cdot4}{6^1^0}=\dfrac{4}{6^{10-6}}=\\\\\\=\dfrac{4}{6^4}=\dfrac{4}{1296}=\dfrac{1}{324}[/tex]
[tex]d)\ \ \dfrac{0,25^3:0,5^3}{5^3}=\dfrac{(0,25:0,5)^3}{5^3}=\dfrac{0,5^3}{5^3}=(0,5:5)^3=0,1^3=0,001\\\\\\e)\ \ \dfrac{(8^5\cdot4^3)^2\cdot25^2}{10^4\cdot2^1^0}=\dfrac{(8^5)^2\cdot(4^3)^2\cdot(5^2)^2}{(5\cdot2)^4\cdot2^1^0}=\dfrac{8^1^0\cdot4^6\cdot\not5^4}{\not5^4\cdot2^4\cdot2^1^0}=\dfrac{(2^3)^1^0\cdot(2^2)^6}{2^{4+10}}=\\\\\\=\dfrac{2^3^0\cdot2^1^2}{2^1^4}=\dfrac{2^{30+12}}{2^1^4}=\dfrac{2^4^2}{2^1^4}=2^{42-14}=2^{28}[/tex]
[tex]f)\ \ \dfrac{64^2\cdot36^4}{6^3\cdot2^6}=\dfrac{(2^6)^2\cdot(6^2)^4}{6^3\cdot2^6}=\dfrac{2^1^2\cdot6^8}{6^3\cdot2^6}=2^{12-6}\cdot6^{8-3}=2^6\cdot6^5=2\cdot2^5\cdot6^5 =\\\\\\=2\cdot(2\cdot6)^5=2\cdot12^5\\\\\\Zastosowane\ \ wzory\\\\a^m\cdot a^n=a^{m+n}\\\\a^m:a^n=a^{m-n}\\\\a^n\cdot b^n=(a\cdot b)^n\\\\(a^m)^n=a^{m\cdot n}[/tex]