Uprość wyrażenie: a. 125*15/3³*5⁴ b. (3⁵)⁴*6⁴/9⁷*4² c. 9³*4⁴/6¹⁰ d. 0,25³ : 0,5³/5³ e. (8⁵*4³)²*25²/10⁴*2¹⁰ f. 64²*36⁴/6³*2⁶​.

[tex]a) \ \frac{125\cdot15}{3^{3}\cdot5^{4}} = \frac{5^{3}\cdot5\cdot3}{3^{3}\cdot5^{4}} = \frac{5^{4}\cdot3^{1}}{5^{4}\cdot3^{3}} =\frac{1}{3^{2}} = \underline{\frac{1}{9}}[/tex]

[tex]b) \ \frac{(3^{5})^{4}\cdot6^{4}}{9^{7}\cdot4^{2}} = \frac{3^{20}\cdot(3\cdot2)^{4}}{(3^{2})^{7}\cdot(2^{2})^{2}} = \frac{3^{20}\cdot3^{4}\cdot2^{4}}{3^{14}\cdot2^{4}} = \frac{3^{24}\cdot2^{4}}{3^{14}\cdot2^{4}} = 3^{24-14} = \underline{3^{10}}[/tex]

[tex]c) \ \frac{9^{3}\cdot4^{4}}{6^{10}} = \frac{(2^{2})^{3}\cdot(2^{2})^{4}}{(3\cdot2)^{10}} = \frac{3^{6}\cdot2^{8}}{3^{10}\cdot2^{10}} = \underline{\frac{1}{3^{4}\cdot2^{2}}}[/tex]

[tex]d) \ \frac{0,25^{3}:0,5^{3}}{5^{3}} = (\frac{0,25:0,5}{5})^{3} = (\frac{0,5}{5})^{3} = 0,1^{3}=\underline{0,001}[/tex]

[tex]e) \ \frac{(8^{5}\cdot4^{3})^{2}\cdot25^{2}}{10^{4}\cdot2^{10}} = \frac{((2^{3})^{5}\cdot(2^{2})^{4}\cdot(5^{2})^{2}}{(2\cdot5)^{4}\cdot2^{10}} = \frac{(2^{15}\cdot2^{6})^{2}\cdot5^{4}}{2^{4}\cdot5^{4}\cdot2^{10}}=\frac{(2^{21})^{2}\cdot5^{4}}{2^{14}\cdot5^{4}} =\frac{2^{42}}{2^{14}} = \underline{2^{28}}[/tex]

[tex]f) \ \frac{64^{2}\cdot36^{4}}{6^{3}\cdot2^{6}} =\frac{(2^{6})^{2}\cdot(6^{2})^{4}}{2^{6}\cdot6^{3}} = \frac{2^{12}\cdot6^{8}}{2^{6}\cdot6^{3}} = 2^{6}\cdot6^{5}=2^{6}\cdot(2\cdot3)^{5} = 2^{6}\cdot2^{5}\cdot3^{5} = \underline{2^{11}\cdot3^{5}}[/tex]

Wyjaśnienie

Wykorzystano własności potęg:

[tex]a^{n}\cdot a^{m} = a^{n+m}\\\\a^{n}:a^{m} = a^{n-m}\\\\(a\cdot b)^{n} = a^{n}\cdot b^{n}\\\\(a^{n})^{m} = a^{n\cdot m}[/tex]

Cześć!

Szczegółowe wyjaśnienie:

Przykład a)

[tex] \frac{125 \cdot15}{3 {}^{3} \: \cdot \: 5 {}^{4} } = \frac{5 {}^{3} \cdot15}{3 {}^{ 3} \cdot5 {}^{4} } = \frac{15 }{3 {}^{3} \cdot5 {}^{} } = \frac{3}{3 {}^{3} } = \frac{1}{3 {}^{2} } = \frac{1}{9} [/tex]

Przykład b)

[tex] \frac{(3 {}^{5}) {}^{4} \cdot6 {}^{4} }{9 {}^{7} \cdot4 {}^{2} } = \frac{3 {}^{20} \cdot6 {}^{4} }{3 {}^{14} \cdot16} = \frac{3 {}^{6} \cdot6 {}^{4} }{16} = \frac{3 {}^{2} \cdot18 {}^{4} }{16} = \frac{9 \cdot18 {}^{4} }{16} [/tex]

Przykład c)

[tex] \frac{9 {}^{3} \cdot4 {}^{4} }{6 {}^{10} } = \frac{4 \cdot36 {}^{3} }{6 {}^{10}} = \frac{4 \cdot6 {}^{6} }{ 6 {}^{10} } = \frac{4}{6 {}^{4} } = \frac{4}{1296} \ | ( : 4 )= \frac{1}{324} [/tex]

Przykład d)

[tex] \frac{0.25 {}^{3} \cdot0.5 {}^{3} }{5 {}^{3} } = \frac{(0.25 \cdot0.5 {}^{3} )}{5 {}^{3} } = \frac{0.125 {}^{3} }{5 {}^{3} } = \frac{ (\frac{1}{8}) {}^{3} }{5 {}^{3} } = \frac{ \frac{1}{512} }{5 {}^{3} } = \frac{1}{512 \cdot5 {}^{ 3} } = \frac{1}{512 \cdot125} = \frac{1}{64000} [/tex]

Przykład e)

[tex] \frac{(8 {}^{5}\cdot4 {}^{3} ) {}^{2} \cdot25 {}^{2} }{10 {}^{4} } = \frac{(2 {}^{15} \cdot2 {}^{6} ) {}^{2} \cdot5 {}^{4} }{5 {}^{4} \cdot2 {}^{4} } = \frac{(2 {}^{21} {}^{} ) {}^{2} }{2 {}^{4} } = \frac{2 {}^{42} }{2 {}^{4} } = 2 {}^{42 - 4} = 2 {}^{38} [/tex]

Przykład f)

[tex] \frac{64 {}^{2} \cdot36 {}^{4} = }{6 {}^{3} \cdot2 {}^{6} = } \frac{2 {}^{12} \cdot6 {}^{8} }{6 {}^{3} \cdot2 {}^{6} } = 2 {}^{6} \cdot6 {}^{5} = 2 \cdot12 {}^{5} [/tex]