Korzystamy ze wzorów:
[tex]zad.~~9.1.\\a)\\\\\sqrt[5]{-32} =\sqrt[5]{\left(-2\right)^{5} }=-2 \\\\\sqrt[4]{\dfrac{1}{81} } =\sqrt[4]{\left(\dfrac{1}{3} \right)^{4} } =\dfrac{1}{3} \\\\\sqrt[6]{64}=\sqrt[6]{2^{6} } =2\\\\\sqrt[10]{1024} =\sqrt[10]{2^{10} } =2\\\\b)\\\\\sqrt[5]{\dfrac{1}{32} } =\sqrt[5]{\left(\dfrac{1}{2} \right)^{5} } =\dfrac{1}{2}\\\\\sqrt[4]{625} =\sqrt[4]{5^{4} } =5\\\\[/tex]
[tex]\sqrt[7]{\dfrac{128}{2187} } =\sqrt[7]{\left(\dfrac{2}{3} \right)^{7} } =\dfrac{2}{3}\\\\\sqrt[5]{-\dfrac{243}{1024} } =\sqrt[5]{\left(-\dfrac{3}{4} \right)^{5} } =-\dfrac{3}{4}[/tex]
[tex]c)\\\\\sqrt[11]{\dfrac{1}{2048} } =\sqrt[11]{\left(\dfrac{1}{2} \right)^{11} } =\dfrac{1}{2}\\\\\sqrt[6]{0,000064} =\sqrt[6]{\dfrac{64}{1000000} } =\sqrt[6]{\left(\dfrac{2}{10} \right)^{6} } =\dfrac{2}{10}=\dfrac{1}{5} \\\\\sqrt[5]{0,00243} =\sqrt[5]{\dfrac{243}{100000} } =\sqrt[5]{\left(\dfrac{3}{10} \right)^{5} } =\dfrac{3}{10}=0,3\\\\\sqrt[4]{\dfrac{16}{625} } =\sqrt[4]{\left(\dfrac{2}{5} \right)^{4} } =\dfrac{2}{5}[/tex]
[tex]zad.~~9.2.\\\\a)\\\\\sqrt[3]{216} =\sqrt[3]{6^{3} } =6\\\\\sqrt[6]{729} =\sqrt[6]{3^{6} } =3\\\\\sqrt[3]{3\dfrac{3}{8} } =\sqrt[3]{\dfrac{27}{8} } =\sqrt[3]{\left(\dfrac{3}{2} \right)^{3} } =\dfrac{3}{2} =1\dfrac{1}{2} \\\\\sqrt[4]{625} =\sqrt[4]{5^{4} } =5\\\\\sqrt[8]{256} =\sqrt[8]{2^{8} } =2\\\\\boxed{\sqrt[3]{3\dfrac{3}{8} } ~~ < ~~\sqrt[8]{256} ~~ < ~~\sqrt[6]{729} ~~ < ~~\sqrt[4]{625}~~ < ~~\sqrt[3]{216} }[/tex]
[tex]b)\\\\\sqrt[5]{243} =\sqrt[5]{5^{5} } =5\\\\\sqrt[3]{1\dfrac{91}{125} } =\sqrt[3]{\dfrac{216}{125} } =\sqrt[3]{\left(\dfrac{6}{5} \right)^{3} } =\dfrac{6}{5} =1\dfrac{1}{5}=1,2 \\\\\sqrt[4]{0,0081} =\sqrt[4]{\dfrac{81}{10000} } =\sqrt[4]{\left(\dfrac{3}{10} \right)^{4} } =\dfrac{3}{10} =0,3\\\\\sqrt[3]{-216} =\sqrt[3]{\left(-6\right)^{3} } =-6\\\\[/tex]
[tex]\sqrt[4]{5\dfrac{1}{16} } =\sqrt[4]{\dfrac{81}{16} } =\sqrt[4]{\left(\dfrac{3}{2} \right)^{4} } =\dfrac{3}{2} =1\dfrac{1}{2}=1,5\\\\\boxed{\sqrt[3]{-216}~~ < ~~\sqrt[4]{0,0081} ~~ < ~~\sqrt[3]{1\dfrac{91}{125} } ~~ < ~~\sqrt[4]{5\dfrac{1}{16} }~~ < ~~\sqrt[5]{243}}[/tex]
[tex]zad.~~9.3.\\\\a)\\\\\sqrt[4]{162} =\sqrt[4]{81\cdot 2}=\sqrt[4]{81} \cdot \sqrt[4]{2} =\sqrt[4]{3^{4} } \cdot \sqrt[4]{2}=3\sqrt[4]{2} \\\\\sqrt[5]{-160} =\sqrt[5]{\left(-32\right)\cdot 5}=\sqrt[5]{-32} \cdot \sqrt[5]{5} =\sqrt[5]{\left(-2\right)^{5} } \cdot \sqrt[5]{5}=-2\sqrt[5]{5}\\\\[/tex]
[tex]\sqrt[4]{0,0048} =\sqrt[4]{\dfrac{48}{10000} } =\sqrt[4]{\dfrac{16\cdot 3}{10^{4} } } =\sqrt[4]{\dfrac{2^{4} \cdot 3}{10^{4} } } =\sqrt[4]{\left(\dfrac{2}{10} \right)^{4} } \cdot \sqrt[4]{3}= \dfrac{2}{10} \cdot \sqrt[4]{3}=\dfrac{\sqrt[4]{3} }{5}[/tex]
[tex]b)\\\\\sqrt[5]{-\dfrac{243}{256} } =\sqrt[5]{-\dfrac{3^{5} }{32\cdot 8} } =\sqrt[5]{-\dfrac{3^{5} }{2^{5} \cdot 8} }=\sqrt[5]{\left(-\dfrac{3}{2} \right)^{2} } \cdot \dfrac{1}{\sqrt[5]{8} } =-\dfrac{3}{2} \cdot \dfrac{1}{\sqrt[5]{8} } \cdot \dfrac{\sqrt[5]{4} }{\sqrt[5]{4} } =- \dfrac{3\sqrt[5]{4} }{2\sqrt[5]{8\cdot 4} } =- \dfrac{3\sqrt[5]{4} }{2\sqrt[5]{32} }=- \dfrac{3\sqrt[5]{8} }{2\sqrt[5]{2^{5} } }=- \dfrac{3\sqrt[5]{8} }{2\cdot 2 }=- \dfrac{3\sqrt[5]{8} }{4 }[/tex]
[tex]\sqrt[4]{405} =\sqrt[4]{81\cdot 5} =\sqrt[4]{81}\cdot \sqrt[4]{ 5}=\sqrt[4]{3^{4} }\cdot \sqrt[4]{ 5}= 3\sqrt[4]{ 5}\\\\\sqrt[5]{192} =\sqrt[5]{32\cdot 6} =\sqrt[5]{32} \cdot \sqrt[5]{6} =\sqrt[5]{2^{5} } \cdot \sqrt[5]{6} =2\sqrt[5]{6}[/tex]
[tex]c)\\\\\sqrt[3]{-648} =\sqrt[3]{\left(-216\right)\cdot 3} =\sqrt[3]{\left(-6\right)^{3} } \cdot \sqrt[3]{3} =-6\sqrt[3]{3} \\\\\sqrt[5]{486} =\sqrt[5]{243\cdot 2} =\sqrt[5]{243} \cdot \sqrt[5]{2} =\sqrt[5]{3^{5} } \cdot \sqrt[5]{2} =3\sqrt[5]{2} \\\\\sqrt[7]{512} =\sqrt[7]{128\cdot 4}=\sqrt[7]{128}\cdot \sqrt[7]{4}=\sqrt[7]{2^{7} }\cdot \sqrt[7]{4}=2\sqrt[7]{4}[/tex]
