Esercizi sulla razionalizzazione

Esercizio

Razionalizza i denominatori delle seguenti frazioni.

1) \frac{2}{\sqrt{3}} ; \frac{7}{2\sqrt{5}}\frac{3\sqrt{5}}{4\sqrt{3}}.

2)\frac{2\sqrt{5}-3\sqrt{2}}{2\sqrt{10}}\frac{\sqrt{3}-1}{3\sqrt{2}}\frac{1}{3\sqrt{a+b}}

3)\frac{2}{\sqrt[3]{2}}\frac{5}{2\sqrt[4]{8}}\frac{3\sqrt[3]{3}}{2\sqrt[6]{32}}

4)\frac{\sqrt{5}}{3\sqrt{2}-2\sqrt{3}} 

5)\frac{2\sqrt{6}}{7 +3\sqrt{3}}

6)\frac{2\sqrt{3}+1}{2\sqrt{5} -3\sqrt{2}}

7)\frac{a}{\sqrt{a}+\sqrt{b}}

SVOLGIMENTO

1) \frac{2}{\sqrt{3}} ; \frac{7}{2\sqrt{5}}\frac{3\sqrt{5}}{4\sqrt{3}}.

  • \frac{2}{\sqrt{3}} = \frac{2 \cdot\sqrt{3}}{\sqrt{3} \cdot\sqrt{3} } = \frac{2 \sqrt{3}}{3 } 
  • \frac{7}{2\sqrt{5}} = \frac{7 \cdot \sqrt{5}}{2\sqrt{5} \cdot\sqrt{5} } = \frac{7 \cdot \sqrt{5}}{2\cdot5 } = \frac{7 }{10 }\sqrt{5}
  •  \frac{3\sqrt{5}}{4\sqrt{3}} =  \frac{3\sqrt{5}\cdot \sqrt{3}}{4\sqrt{3} \cdot \sqrt{3}} =  \frac{3\sqrt{15}}{4\cdot 3} =  \frac{\sqrt{15}}{4}

2)\frac{2\sqrt{5}-3\sqrt{2}}{2\sqrt{10}}\frac{\sqrt{3}-1}{3\sqrt{2}}\frac{1}{3\sqrt{a+b}}

  • \frac{2\sqrt{5}-3\sqrt{2}}{2\sqrt{10}} = \frac{(2\sqrt{5}-3\sqrt{2})\sqrt{10  }}{2\sqrt{10 \cdot }\sqrt{10  }} = \frac{(2\sqrt{50}-3\sqrt{20})}{2\cdot 10} = \frac{(2\sqrt{5 ^{2}\cdot2}-3\sqrt{2 ^{2}\cdot5})}{20}=

\frac{(2\cdot5\sqrt{2}-3\cdot2\sqrt{5})}{20}\frac{2(\cdot5\sqrt{2}-3\sqrt{5})}{20}\frac{5\sqrt{2}-3\sqrt{5}}{10}

  • \frac{\sqrt{3}-1}{3\sqrt{2}}\frac{(\sqrt{3}-1)\cdot \sqrt{2}}{3\sqrt{2}\cdot \sqrt{2}} = \frac{(\sqrt{6}- \sqrt{2})}{3\cdot 2} = \frac{(\sqrt{6}- \sqrt{2})}{6}
  • \frac{1}{3\sqrt{a+b}} = \frac{1 \cdot\sqrt{a+b}}{3\sqrt{a+b}\cdot\sqrt{a+b}} = \frac{\sqrt{a+b}}{3(a+b)}

3)\frac{2}{\sqrt[3]{2}}\frac{5}{2\sqrt[4]{8}}\frac{3\sqrt[3]{3}}{2\sqrt[6]{32}}

  • \frac{2}{\sqrt[3]{2}}= \frac{2\cdot\sqrt[3]{2^{2}}}{\sqrt[3]{2}\cdot\sqrt[3]{2^{2}} } = \frac{2\cdot\sqrt[3]{2^{2}}}{\sqrt[3]{2^{3}} } = \frac{2\cdot\sqrt[3]{2^{2}}}{2 } = \sqrt[3]{4}
  • \frac{5}{2\sqrt[4]{8}} = \frac{5}{2\sqrt[4]{2 ^{3}}} = \frac{5\sqrt[4]{2 }}{2\sqrt[4]{2 ^{3}}\cdot\sqrt[4]{2 }}  = \frac{5\sqrt[4]{2 }}{2\sqrt[4]{2 ^{4}}}  = \frac{5\sqrt[4]{2 }}{2\cdot 2}}  = \frac{5\sqrt[4]{2 }}{4}}
  • \frac{3\sqrt[3]{3}}{2\sqrt[6]{32}}\frac{3\sqrt[3]{3}}{2\sqrt[6]{2 ^{5}}} = \frac{3\sqrt[3]{3}\cdot\sqrt[6]{2 }  }{2\sqrt[6]{2 ^{5}}\cdot\sqrt[6]{2 }  = \frac{3\sqrt[3]{3}\cdot\sqrt[6]{2 }  }{2\sqrt[6]{2 ^{6}} = \frac{3\sqrt[6]{3 ^{2}}\cdot\sqrt[6]{2 }  }{2\cdot 2}  ho fatto diventare le radici del numeratore dello stesso indice in modo da poterle moltiplicare = \frac{3\sqrt[6]{18}  }{4}

4)\frac{\sqrt{5}}{3\sqrt{2}-2\sqrt{3}} = \frac{\sqrt{5}(3\sqrt{2}+2\sqrt{3})}{(3\sqrt{2}-2\sqrt{3})(3\sqrt{2}+2\sqrt{3})} = \frac{(3\sqrt{10}+2\sqrt{15})}{9 \cdot2-4\cdot3}  = \frac{3\sqrt{10}+2\sqrt{15}}{18-12}  = \frac{(3\sqrt{10}+2\sqrt{15})}{6}

5)\frac{2\sqrt{6}}{7 +3\sqrt{3}} = \frac{2\sqrt{6}(7 -3\sqrt{3})}{(7 +3\sqrt{3})(7 -3\sqrt{3})} = \frac{14\sqrt{6}-6\sqrt{18})}{(49 -27)} = \frac{14\sqrt{6}-18\sqrt{2}}{22}  \frac{2(7\sqrt{6}-9\sqrt{2})}{22}  \frac{7\sqrt{6}-9\sqrt{2}}{11}

6)\frac{2\sqrt{3}+1}{2\sqrt{5} -3\sqrt{2}} =\frac{(2\sqrt{3}+1)(2\sqrt{5} +3\sqrt{2})}{(2\sqrt{5} -3\sqrt{2})(2\sqrt{5} +3\sqrt{2})} = \frac{4\sqrt{15}+6\sqrt{6}+2\sqrt{5} +3\sqrt{2}}{(20 -18)} =\frac{4\sqrt{15}+6\sqrt{6}+2\sqrt{5} +3\sqrt{2}}{2}

7)\frac{a}{\sqrt{a}+\sqrt{b}} = \frac{a(\sqrt{a}-\sqrt{b})}{(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})}  \frac{a(\sqrt{a}-\sqrt{b})}{a-b}

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