Odpowiedź :
Odpowiedź:
[tex]\huge\boxed{~~a)~~\dfrac{\frac{8^{-6}}{16^{-3}} }{4^{7}} =\dfrac{1}{1~048~576} ~~}[/tex]
[tex]\huge\boxed{~~b)~~\left(\dfrac{2}{3} \right)^{-3} \cdot \left(\dfrac{27}{8} \right)^{-3} \cdot (1,5)^{-3}=\dfrac{512}{19~~683} ~~}[/tex]
[tex]\huge\boxed{~~c)~~log_\frac{1}{6} \dfrac{1}{216} +log_{0,25} 16=1~~}[/tex]
[tex]\huge\boxed{ ~~d)~~log_{1,5}\dfrac{9}{4} -log_{\pi }1= 2~~ }[/tex]
[tex]\huge\boxed{~~e)~~log_{3}\dfrac{1}{12} +log_{3}\dfrac{14}{15} + log_{3}\dfrac{10}{21} = -2~~}[/tex]
[tex]\huge\boxed{~~f)~~3^{\frac{2}{3} } \cdot 3^{-\frac{4}{3} } \cdot 9^{2}=27 \sqrt[3]{3} ~~}[/tex]
Szczegółowe wyjaśnienie:
Korzystamy ze wzorów:
- [tex]log_{a} b+log_{a}c=log_{a} (b\cdot c),~~zal.~~a > 0,~~a\neq 1,~~b > 0,~~c > 0[/tex]
- [tex]log_{a} b-log_{a}c=log_{a} (b\div c),~~zal.~~a > 0,~~a\neq 1,~~b > 0,~~c > 0[/tex]
- [tex]log_{a}1=0[/tex]
- [tex]log_{a} b^{k}=k\cdot log_{a}b,~~zal.~~a > 0,~~a\neq 1,~~b > 0[/tex]
- [tex]log_{a} a=1,~~zal.~~a > 0,~~a\neq 1[/tex]
- [tex]x^{n} \cdot x^{m} =x^{n+m}[/tex]
- [tex]x^{n} \div x^{m} =x^{n-m}[/tex]
- [tex]\left( \dfrac{x}{y} \right)^{n}=\left( \dfrac{y}{x} \right)^{-n},~~zal.~~x > 0,~~y > 0[/tex]
- [tex](x^{n})^{m} =x^{n\cdot m }[/tex]
- [tex]\sqrt[n]{x^{n}}} =x^{n\cdot \frac{1}{n} }=x^{1}=x,~~zal.~~x\geq 0[/tex]
- [tex]\sqrt[n]{x} =x^{\frac{1}{n} } ,~~zal.~~x\geq 0[/tex]
Obliczamy:
[tex]\boxed{a)}\\\\\dfrac{\frac{8^{-6}}{16^{-3}} }{4^{7}} =\dfrac{\frac{(2^{3})^{-6}}{(2^{4})^{-3}} }{(2^{2})^{7}} =\dfrac{\frac{2^{-18}}{2^{-12}} }{2^{14}} =\dfrac{2^{-18-(-12)}}{2^{14}} =\dfrac{2^{-18+12}}{2^{14}} =\dfrac{2^{-6}}{2^{14}} =2^{-6-14}=2^{-20}=\dfrac{1}{2^{20}} =\boxed{\dfrac{1}{1~048~576} }[/tex]
[tex]\boxed{b)}\\\\\left(\dfrac{2}{3} \right)^{-3} \cdot \left(\dfrac{27}{8} \right)^{-3} \cdot (1,5)^{-3}=\left(\dfrac{2\!\!\!\!\diagup^1}{3\!\!\!\!\diagup_1} \cdot \dfrac{27}{8} \cdot \dfrac{3\!\!\!\!\diagup^1}{2\!\!\!\!\diagup_1} \right)^{-3} =\left(\dfrac{27}{8} \right)^{-3} =\left(\dfrac{8}{27} \right)^{3} =\dfrac{8\cdot 8\cdot 8}{27\cdot 27 \cdot 27} =\boxed{\dfrac{512}{19~683} }[/tex]
[tex]\boxed{c)}\\\\log_\frac{1}{6} \dfrac{1}{216} +log_{0,25} 16=log_\frac{1}{6} \left(\dfrac{1}{6} \right)^{3} +log_{\frac{1}{4} } 4^{2}=3\cdot log_\frac{1}{6} \dfrac{1}{6}+ log_{\frac{1}{4} } \left( \frac{1}{4} \right)^{-2}=3\cdot 1 + (-2)\cdot log_{\frac{1}{4} }\dfrac{1}{4} =3-2\cdot 1 =3-2=\boxed{1}[/tex]
[tex]\boxed{d)}\\\\log_{1,5} }\dfrac{9}{4} -log_{\pi }1=log_{\frac{3}{2} } } \left(\dfrac{3}{2} \right)^{2} -0=2\cdot log_{\frac{3}{2} } }\dfrac{3}{2}=2\cdot 1 = \boxed{2}[/tex]
[tex]\boxed{e)}\\\\log_{3}\dfrac{1}{12} +log_{3}\dfrac{14}{15} + log_{3}\dfrac{10}{21} = log_{3} \left(\dfrac{1}{12} \cdot \dfrac{14\!\!\!\!\!\diagup^2}{15\!\!\!\!\!\diagup_3} \cdot \dfrac{10\!\!\!\!\!\diagup^2}{21\!\!\!\!\!\diagup_3} \right)=log_{3} \dfrac{4\!\!\!\!\diagup^1}{12\!\!\!\!\!\diagup_3 \cdot 3} =log_{3} \dfrac{1}{9} =log_{3}9^{-1} =log_{3}3^{-2}=-2\cdot log_{3} 3 =-2\cdot 1 = \boxed{-2}[/tex]
[tex]\boxed{f)}\\\\3^{\frac{2}{3} } \cdot 3^{-\frac{4}{3} } \cdot 9^{2}=3^{\frac{2}{3} } \cdot 3^{-\frac{4}{3} } \cdot (3^{2})^{2}=3^{\frac{2}{3} } \cdot 3^{-\frac{4}{3} } \cdot 3^{4}=3^{\frac{2}{3} - \frac{4}{3} +4 } =3^{-\frac{2}{3} +4}=3^{3\frac{1}{3} }=3^{\frac{10}{3} }=3^{10\cdot \frac{1}{3} }=(3^{10})^{\frac{1}{3} } =\sqrt[3]{3^{10}} =\sqrt[3]{3^{3}\cdot 3^{3} \cdot 3^{3} \cdot 3^{1}}=\sqrt[3]{3^{3}} \cdot \sqrt[3]{3^{3}} \cdot \sqrt[3]{3^{3}} \cdot \sqrt[3]{3}=[/tex]
[tex]=3\cdot 3\cdot \cdot 3 \cdot \sqrt[3]{3} =\boxed{27\sqrt[3]{3}}[/tex]