[tex]a)~~\sqrt{2} \cdot (\sqrt{5} +2\sqrt{5} )=\sqrt{2} \cdot 3\sqrt{5}=3\sqrt{2\cdot 5} =3\sqrt{10} \\\\b)~~\dfrac{1}{2} \sqrt[3]{9} \cdot (\sqrt[3]{3} +5\sqrt[3]{3} )=\dfrac{1}{2} \sqrt[3]{9} \cdot 6\sqrt[3]{3}=\dfrac{1}{2}\cdot 6\cdot \sqrt[3]{9} \cdot \sqrt[3]{3}=3\sqrt[3]{9\cdot 3}=3\sqrt[3]{27} =3\sqrt[3]{3^{3} } =3\cdot 3=9\\\\[/tex]
[tex]c)~~7\sqrt{8} \cdot \dfrac{6\sqrt{3} }{\sqrt{6} } =7\sqrt{4\cdot 2} \cdot \dfrac{6\sqrt{3} }{\sqrt{2\cdot 3} }=7\sqrt{4} \cdot \sqrt{2} \cdot \dfrac{6\sqrt{3} }{\sqrt{2} \cdot \sqrt{3} }=7\sqrt{2^{2} } \cdot 6=7\cdot 2\cdot 6 = 84[/tex]
[tex]f)~~\dfrac{\sqrt[3]{24} }{6\sqrt[3]{3} } =\dfrac{\sqrt[3]{8\cdot 3} }{6\sqrt[3]{3} }=\dfrac{\sqrt[3]{8}\cdot \sqrt[3]{3} }{6\sqrt[3]{3} }=\dfrac{\sqrt[3]{8} }{6} =\dfrac{\sqrt[3]{2^{3} } }{6}=\dfrac{2}{6} =\dfrac{1}{3}[/tex]
[tex]g)~~\dfrac{\sqrt[3]{9} \cdot 2\sqrt[3]{6} }{3\sqrt[3]{2} } =\dfrac{\sqrt[3]{9} \cdot 2\sqrt[3]{3\cdot 2} }{3\sqrt[3]{2} }=\dfrac{\sqrt[3]{9} \cdot 2\sqrt[3]{3}\cdot \sqrt[3]{2} }{3\sqrt[3]{2} }=\dfrac{\sqrt[3]{9} \cdot 2\sqrt[3]{3} }{3}=\dfrac{2\sqrt[3]{9\cdot 3} }{3}=\dfrac{2\sqrt[3]{27} }{3}=\dfrac{2\sqrt[3]{3^{3} } }{3}=\dfrac{2\cdot 3}{3} =2[/tex]
[tex]h)~~\dfrac{4\sqrt{6} \cdot 2\sqrt{3} }{3\sqrt{2} } =\dfrac{4\sqrt{2\cdot 3} \cdot 2\sqrt{3} }{3\sqrt{2} } =\dfrac{4\sqrt{3}\cdot \sqrt{2} \cdot 2\sqrt{3} }{3\sqrt{2} } =\dfrac{4\sqrt{3} \cdot 2\sqrt{3} }{3 } =\dfrac{8\sqrt{3\cdot 3} }{3 } =\dfrac{8\sqrt{9} }{3 } =\dfrac{8\sqrt{3^{2} } }{3 } =\dfrac{8\cdot 3}{3} =8[/tex]
Korzystam ze wzorów:
[tex]\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[/tex]
