a)

$\sqrt{41}$  

$8\sqrt{\frac{1}{2}}=\sqrt{64}\cdot \sqrt{\frac{1}{2}}=\sqrt{64\cdot \frac{1}{2}}=\sqrt{32}$  

$5\sqrt{2}=\sqrt{25}\cdot \sqrt{2}=\sqrt{25\cdot 2}=\sqrt{50}$  

$\frac{6}{7}\sqrt{42}=\sqrt{\frac{36}{49}}\cdot \sqrt{42}=\sqrt{\frac{36}{\sout{49}{}_7}\cdot \sout{42}{}^6}=\sqrt{\frac{216}{7}}=\sqrt{30\frac{6}{7}}$  

 

Uporządkujmy liczby rosnąco:

$\sqrt{30\frac{6}{7}}<\sqrt{32}<\sqrt{41}<\sqrt{50}$  

$\frac{6}{7}\sqrt{42}<8\sqrt{\frac{1}{2}}<\sqrt{41}<5\sqrt{2}$  

 

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b)      

$2\sqrt{8}=\sqrt{4}\cdot \sqrt{8}=\sqrt{4\cdot 8}=\sqrt{32}$  

$6\sqrt{2}=\sqrt{36}\cdot \sqrt{2}=\sqrt{36\cdot 2}=\sqrt{72}$  

$\sqrt{40}$  

$3\sqrt{6}=\sqrt{9}\cdot \sqrt{6}=\sqrt{9\cdot 6}=\sqrt{54}$  

 

$\sqrt{32}<\sqrt{40}<\sqrt{54}<\sqrt{72}$  

$2\sqrt{8}<\sqrt{40}<3\sqrt{6}<\sqrt{72}$  

 

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c)

$\sqrt[3]{0,8}$  

$\frac{1}{4}\sqrt[3]{192}=\sqrt[3]{\frac{1}{64}}\cdot \sqrt[3]{192}=\sqrt[3]{\frac{1}{\sout{64}{}_1}\cdot \sout{192}{}^3}=\sqrt[3]{3}$  

$\frac{1}{3}\sqrt[3]{\frac{2}{7}}=\sqrt[3]{\frac{1}{27}}\cdot \sqrt[3]{\frac{2}{7}}=\sqrt[3]{\frac{1}{27}\cdot \frac{2}{7}}=\sqrt[3]{\frac{2}{189}}$  

$3\sqrt[3]{2}=\sqrt[3]{27}\cdot \sqrt[3]{2}=\sqrt[3]{27\cdot 2}=\sqrt[3]{54}$  

 

$\sqrt[3]{\frac{2}{189}}<\sqrt[3]{0,8}<\sqrt[3]{3}<\sqrt[3]{54}$  

$\frac{1}{3}\sqrt[3]{\frac{2}{7}}<\sqrt[3]{0,8}<\frac{1}{4}\sqrt[3]{192}<3\sqrt[3]{2}$  

 

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d)

$5\sqrt[3]{0,2}=\sqrt[3]{125}\cdot \sqrt[3]{0,2}=\sqrt[3]{125\cdot 0,2}=\sqrt[3]{25}$  

$2\sqrt[3]{3}=\sqrt[3]{8}\cdot \sqrt[3]{3}=\sqrt[3]{8\cdot 3}=\sqrt[3]{24}$  

$\frac{1}{2}\sqrt[3]{320}=\sqrt[3]{\frac{1}{8}}\cdot \sqrt[3]{320}=\sqrt[3]{\frac{1}{8}\cdot 320}=\sqrt[3]{40}$  

$\sqrt[3]{30}$  

 

$\sqrt[3]{24}<\sqrt[3]{25}<\sqrt[3]{30}<\sqrt[3]{40}$  

$2\sqrt[3]{3}<5\sqrt[3]{0,2}<\sqrt[3]{30}<\frac{1}{2}\sqrt[3]{320}$