Korzystamy ze wzorów:
[tex]\sqrt[n]{x} =x^{\frac{1}{n} } \\\\\sqrt[n]{x} \cdot \sqrt[n]{y} =\sqrt[n]{x\cdot y} \\\\\sqrt[n]{x^{n} } =x^{n\cdot \frac{1}{n} } =x^{1} =x\\\\\sqrt[n]{x} \div \sqrt[n]{y} =\sqrt[n]{x\div y}=\sqrt[n]{x\cdot \dfrac{1}{y} }=\sqrt[n]{\dfrac{x}{y} }[/tex]
[tex]\boxed{zad.a}\\\\3\sqrt{2} -\sqrt{12,5} =3\sqrt{2} -\sqrt{12\frac{1}{2} }=3\sqrt{2} -\sqrt{\frac{25}{2} }=3\sqrt{2} -\sqrt{\frac{5^{2} }{2} }=3\sqrt{2} -\frac{5}{\sqrt{2} } \cdot \frac{\sqrt{2} }{\sqrt{2} } =3\sqrt{2} -\frac{5\sqrt{2} }{2} =\frac{6\sqrt{2} }{2} -\frac{5\sqrt{2} }{2} =\frac{6\sqrt{2} -5\sqrt{2} }{2} =\boxed{\frac{\sqrt{2} }{2}}[/tex]
[tex]\boxed{zad.b}\\\\\sqrt[3]{7} \cdot 3^{3} \sqrt{4^{9} } =\sqrt[3]{7} \cdot 27\cdot \sqrt{(2^{2} )^{9} } =27\sqrt[3]{7} \cdot \sqrt{2^{2\cdot 9} } =27\sqrt[3]{7} \cdot \sqrt{2^{18} } =27\sqrt[3]{7} \cdot (2^{18} )^{\frac{1}{2} } =27\sqrt[3]{7} \cdot 2^{18\cdot \frac{1}{2} }=27\sqrt[3]{7} \cdot 2^{9}=27\sqrt[3]{7} \cdot 512=\boxed{13~824\sqrt[3]{7}}[/tex]
[tex]\boxed{zad.c}\\\\5\sqrt{125} \div 2\sqrt{6} =5\sqrt{25\cdot 5} \div 2\sqrt{6} =5\sqrt{5^{2} \cdot 5} \div 2\sqrt{6}=25\sqrt{5} \div 2\sqrt{6} =\dfrac{25\sqrt{5} }{2\sqrt{6} } \cdot \dfrac{\sqrt{6} }{\sqrt{6} } =\dfrac{25\sqrt{5\cdot 6} }{2\sqrt{6\cdot 6} }=\dfrac{25\sqrt{30} }{2\sqrt{6^{2} } }=\dfrac{25\sqrt{30} }{2\cdot 6 }=\dfrac{25\sqrt{30} }{12 }=\boxed{\dfrac{25}{12} \sqrt{30}}[/tex]
[tex]\boxed{zad.d}\\\\\sqrt{9} \div 2\sqrt{4} \cdot \sqrt{2} =\sqrt{3^{2} } \div 2\sqrt{2^{2} } \cdot \sqrt{2} =3\div 4\cdot \sqrt{2} =3\cdot \dfrac{1}{4} \cdot \sqrt{2} = \dfrac{3\sqrt{2} }{4} =\boxed{ \dfrac{3}{4} \sqrt{2}}[/tex]
[tex]\boxed{zad.e}\\\\\sqrt[3]{4} \div \sqrt[3]{9} \cdot \sqrt[3]{6} =\sqrt[3]{4\div 9\cdot 6} =\sqrt[3]{4\cdot \dfrac{1}{9} \cdot 6} =\sqrt[3]{4\cdot \dfrac{1}{3} \cdot 2} =\sqrt[3]{\dfrac{8}{3} } =\sqrt[3]{\dfrac{2^{3} }{3} }=\dfrac{2}{\sqrt[3]{3} } \cdot \dfrac{\sqrt[3]{9} }{\sqrt[3]{9} } =\dfrac{2\sqrt[3]{9} }{\sqrt[3]{3\cdot 9} } =\dfrac{2\sqrt[3]{9} }{\sqrt[3]{3^{3} } } =\dfrac{2\sqrt[3]{9} }{3} =\boxed{\dfrac{2}{3} \sqrt[3]{9}}[/tex]
[tex]\boxed{zad.f}\\\\(4\sqrt{50^{5} }) \cdot (2\sqrt{2} )=(4\sqrt{(25\cdot 2)^{5} }) \cdot (2\sqrt{2} )=(4\sqrt{(5^{2} \cdot 2)^{5} }) \cdot (2\sqrt{2} )=(4\sqrt{5^{2\cdot 5} \cdot 2^{5} } )\cdot (2\sqrt{2} )=(4\sqrt{5^{10} \cdot 2^{5} } )\cdot (2\sqrt{2} )=8\cdot \sqrt{5^{10} } \cdot \sqrt{2^{5} } \cdot \sqrt{2} =8\cdot 5^{10\cdot \frac{1}{2} } \cdot \sqrt{2^{5} \cdot 2} =[/tex]
[tex]=8\cdot 5^{5} \cdot \sqrt{2^{5+1} } =8\cdot 3~125\cdot \sqrt{2^{6} } =25~000\cdot 2^{6\cdot \frac{1}{2} } =25~000\cdot 2^{3} =25~000\cdot 8=\boxed{200000}[/tex]