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Potęgi. Ćwiczenia - cz.5.

śr., 09/15/2010 - 15:05 — ciebie

Ćwiczenie 5.1 Oblicz:

  1. \(\displaystyle{\sqrt[3]{-8}}\;\),
  2. \(\displaystyle{\sqrt[3]{-27}}\;\),
  3. \(\displaystyle{\sqrt[5]{-32}}\;\),
  4. \(\displaystyle{\sqrt[3]{-\frac{1}{64}}}\;\),
  5. \(\displaystyle{\sqrt[3]{-\frac{1}{125}}}\;\),
  6. \(\displaystyle{\sqrt[7]{-\frac{1}{128}}}\;\),
  7. \(\displaystyle{\sqrt[3]{-0,216}}\;\),
  8. \(\displaystyle{\sqrt[3]{-0,000008}}\;\),
  9. \(\displaystyle{\sqrt[5]{-0,00243}}\;\).


Odpowiedź:

  1. \(\displaystyle{-2}\;\).
  2. \(\displaystyle{-3}\;\).
  3. \(\displaystyle{-2}\;\).
  4. \(\displaystyle{-\frac{1}{4}}\;\).
  5. \(\displaystyle{-\frac{1}{5}}\;\).
  6. \(\displaystyle{-\frac{1}{2}}\;\).
  7. \(\displaystyle{-0,6}\;\).
  8. \(\displaystyle{-0,02}\;\).
  9. \(\displaystyle{-0,3}\;\).


Rozwiązanie:

  1. \(\displaystyle{\sqrt[3]{-8}=\sqrt[3]{-2^3}=-2}\;\).
  2. \(\displaystyle{\sqrt[3]{-27}=\sqrt[3]{-3^3}=-3}\;\).
  3. \(\displaystyle{\sqrt[5]{-32}=\sqrt[5]{-2^5}=-2}\;\).
  4. \(\displaystyle{\sqrt[3]{-\frac{1}{64}}=\sqrt[3]{-\left(\frac{1}{4}\right)^3}=-\frac{1}{4}}\;\).
  5. \(\displaystyle{\sqrt[3]{-\frac{1}{125}}=\sqrt[3]{-\left(\frac{1}{5}\right)^3}=-\frac{1}{5}}\;\).
  6. \(\displaystyle{\sqrt[7]{-\frac{1}{128}}=\sqrt[7]{-\left(\frac{1}{2}\right)^7}=-\frac{1}{2}}\;\).
  7. \(\displaystyle{\sqrt[3]{-0,216}=\sqrt[3]{-(0,6)^3}=-0,6}\;\).
  8. \(\displaystyle{\sqrt[3]{-0,000008}=\sqrt[3]{-(0,02)^3}=-0,02}\;\).
  9. \(\displaystyle{\sqrt[5]{-0,00243}=\sqrt[5]{-(0,3)^5}=-0,3}\;\).
∎

Ćwiczenie 5.2 Oblicz:

  1. \(\displaystyle{\sqrt[5]{(-2)^6\cdot (-3)^{15}}}\;\),
  2. \(\displaystyle{\sqrt[9]{(-5)^{19}\cdot(-2)^{10}\cdot 27^3}}\;\),
  3. \(\displaystyle{ \sqrt[3]{\frac{(-3)^{11}}{8^2}}}\;\),
  4. \(\displaystyle{2\frac{1}{4}\cdot \sqrt[5]{\frac{(-2)^{12}}{(-3)^{11}}}}\;\),
  5. \(\displaystyle{\sqrt[7]{-1^4\cdot\sqrt[5]{(-1)^2\cdot\sqrt[3]{(-1)^3}}}}\;\),
  6. \(\displaystyle{\sqrt[3]{(-5)^6\sqrt[7]{(-3)^{21}\cdot (-8)^{14}}}}\;\),
  7. \(\displaystyle{3\sqrt[3]{-16}+\sqrt[3]{54}-\sqrt[3]{-250}}\;\),
  8. \(\displaystyle{\sqrt[5]{-64}+10\sqrt[5]{0,00002}-2\sqrt[5]{-\frac{1}{16}}}\;\).


Odpowiedź:

  1. \(\displaystyle{-54\sqrt[5]{2}}\;\).
  2. \(\displaystyle{-150\sqrt[9]{10}}\;\).
  3. \(\displaystyle{-\frac{27}{4}\sqrt[3]{9}}\;\).
  4. \(\displaystyle{-\sqrt[5]{\frac{4}{3}}}\;\).
  5. \(1\;\).
  6. \(\displaystyle{-300}\;\).
  7. \(\displaystyle{2\sqrt[3]{2}}\;\).
  8. \(0\;\).


Rozwiązanie:

  1. \(\displaystyle{ \sqrt[5]{(-2)^6\cdot (-3)^{15}}=\sqrt[5]{2^6\cdot (-3^{15})}=\sqrt[5]{2^5\cdot 2 \cdot \left(-(3^3)^5\right)}=2\cdot (-3^3)\cdot\sqrt[5]{2}=-54\sqrt[5]{2}}\;\).
  2. \(\displaystyle{\sqrt[9]{(-5)^{19}\cdot(-2)^{10}\cdot 27^3}=\sqrt[9]{-5^{18}\cdot5\cdot2^9\cdot2\cdot(3^3)^3}= \sqrt[9]{-(5^2)^9\cdot 2^9\cdot3^9\cdot 10}=-5^2\cdot2\cdot3\cdot\sqrt[9]{10}=}\;\)
    \(\displaystyle{-150\sqrt[9]{10}}\;\).
  3. \(\displaystyle{ \sqrt[3]{\frac{(-3)^{11}}{8^2}}=\sqrt[3]{\frac{-3^{11}}{(2^3)^2}}=\sqrt[3]{\frac{-(3^3)^3\cdot 3^2}{(2^2)^3}}= -\frac{3^3}{2^2}\cdot\sqrt[3]{3^2}=-\frac{27}{4}\sqrt[3]{9}}\;\).
  4. \(\displaystyle{\left(2\frac{1}{4}\right)\cdot \sqrt[5]{\frac{(-2)^{12}}{(-3)^{11}}}=\left(2\frac{1}{4}\right)\cdot \sqrt[5]{\frac{2^{12}}{-3^{11}}}= \left(2\frac{1}{4}\right)\cdot \sqrt[5]{\frac{2^{10}\cdot 2^2}{-3^{10}\cdot 3}}=\left(2\frac{1}{4}\right)\cdot \sqrt[5]{-\left(\left(\frac{2}{3}\right)^2\right)^5\cdot\frac{4}{3}}=}\;\)
    \(\displaystyle{ \frac{9}{4}\cdot\left(-\frac{4}{9}\right) \sqrt[5]{\frac{4}{3}}=-\sqrt[5]{\frac{4}{3}}}\;\).
  5. \(\displaystyle{\sqrt[7]{-1^4\cdot\sqrt[5]{(-1)^2\cdot\sqrt[3]{(-1)^3}}}=\sqrt[7]{-\sqrt[5]{1\cdot (-1)}}=\sqrt[7]{-(-1)}=\sqrt[7]{1}=1}\;\).
  6. \(\displaystyle{\sqrt[3]{(-5)^6\cdot\sqrt[7]{(-3)^{21}\cdot (-8)^{14}}} = \sqrt[3]{5^6\cdot\sqrt[7]{-(3^{3})^7\cdot (8^2)^7}}= \sqrt[3]{(5^2)^3\cdot(-3^{3})\cdot (2^3)^2}=}\;\)
    \(\displaystyle{\sqrt[3]{(5^2)^3\cdot(-3^{3})\cdot (2^2)^3}=5^2\cdot (-3)\cdot 2^2=-(2\cdot 5)^2\cdot 3=-10^2\cdot 3=-300 }\;\).
  7. \(\displaystyle{3\sqrt[3]{-16}+\sqrt[3]{54}-\sqrt[3]{-250}=3\sqrt[3]{-8\cdot 2}+\sqrt[3]{27\cdot 2}-\sqrt[3]{-125\cdot 2}= 3\sqrt[3]{-2^3\cdot 2}+\sqrt[3]{3^3\cdot 2}-\sqrt[3]{-5^3\cdot 2}=}\;\)
    \(\displaystyle{-3\cdot 2\sqrt[3]{2}+3\sqrt[3]{2}+5\sqrt[3]{2}=2\sqrt[3]{2}}\;\).
  8. \(\displaystyle{\sqrt[5]{-64}+10\sqrt[5]{0,00002}-2\sqrt[5]{-\frac{1}{16}}=-\sqrt[5]{2^5\cdot 2}+10\sqrt[5]{2\cdot\frac{1}{10^5}}+2\cdot\sqrt[5]{\frac{2}{2^5}}=}\;\)
    \(\displaystyle{-2\sqrt[5]{2}+10\cdot\frac{1}{10}\sqrt[5]{2}+2\cdot\frac{1}{2}\sqrt[5]{2}= -2\sqrt[5]{2}+\sqrt[5]{2}+\sqrt[5]{2}=0}\;\).
∎



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