Dê a expressão mais simples de
[tex]\frac{\frac{\frac{\sqrt[4]{a}}{\sqrt[6]{a}}}{\sqrt[8]{a}}}{\frac{\sqrt[3]{a} . \sqrt[9]{a}}{\sqrt{a}}}[/tex]
[tex]\frac{\frac{\frac{\sqrt[4]{a}}{\sqrt[6]{a}}}{\sqrt[8]{a}}}{\frac{\sqrt[3]{a} . \sqrt[9]{a}}{\sqrt{a}}}[/tex]
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[tex] \frac{ \sqrt[4]{a} }{ \sqrt[6]{a} } = \frac{a ^{ \frac{1}{4} } }{a {}^{ \frac{1}{6} } } = a^{ \frac{1}{4} - \frac{1}{6} } = a^{ \frac{1}{12} } \\ \\ \frac{ \sqrt[12]{a} }{ \sqrt[8]{a} } = \frac{a {}^{ \frac{1}{1 2} } }{a^{ \frac{1}{8} } } = a^{ \frac{1}{12} - \frac{1}{8} } = a {}^{ - \frac{1}{24} } \\ \\ \\ \sqrt[3]{a} . \sqrt[9]{a} = a {}^{\frac{1}{3} } .a {}^{ \frac{1}{9} } = a {}^{ \frac{1}{3} + \frac{1}{9} } = a^{ \frac{4}{9} } \\ \\ \frac{a {}^{ \frac{4}{9} } }{ \sqrt{a} } = \frac{a {}^{ \frac{4}{9} } }{a {}^{ \frac{1}{2} } } = a {}^{ \frac{4}{9} - \frac{1}{2} } = a {}^{ - \frac{1}{18} } \\ \\ \frac{a {}^{ - \frac{1}{24} } }{a {}^{ - \frac{1}{18} } } = a {}^{ - \frac{1}{24} + \frac{1}{18} } = a {}^{ \frac{1}{72} } = \boxed{\sqrt[72]{a} }[/tex]
Resposta:
[tex]\Large \textsf{Leia abaixo}[/tex]
Explicação passo a passo:
[tex]\Large \text{$ \sf \dfrac{\dfrac{\dfrac{\sqrt[\sf 4]{\sf a}}{\sqrt[\sf 6]{\sf a}}}{\sqrt[\sf 8]{\sf a}}}{\dfrac{\sqrt[\sf 3]{\sf a}\:.\:\sqrt[\sf 9]{\sf a}}{\sqrt{\sf a}}} = \dfrac{\dfrac{a^{\frac{1}{4} - \frac{1}{6}}}{\sqrt[\sf 8]{\sf a}}}{\dfrac{\sqrt[\sf 3]{\sf a}\:.\:\sqrt[\sf 9]{\sf a}}{\sqrt{\sf a}}} $}[/tex]
[tex]\Large \text{$ \sf \dfrac{\dfrac{\dfrac{\sqrt[\sf 4]{\sf a}}{\sqrt[\sf 6]{\sf a}}}{\sqrt[\sf 8]{\sf a}}}{\dfrac{\sqrt[\sf 3]{\sf a}\:.\:\sqrt[\sf 9]{\sf a}}{\sqrt{\sf a}}} = \dfrac{a^{\frac{1}{12} - \frac{1}{8}}}{\dfrac{a^{\frac{1}{3} + \frac{1}{9}}}{\sqrt{\sf a}}} $}[/tex]
[tex]\Large \text{$ \sf \dfrac{\dfrac{\dfrac{\sqrt[\sf 4]{\sf a}}{\sqrt[\sf 6]{\sf a}}}{\sqrt[\sf 8]{\sf a}}}{\dfrac{\sqrt[\sf 3]{\sf a}\:.\:\sqrt[\sf 9]{\sf a}}{\sqrt{\sf a}}} = \dfrac{a^{\frac{1}{12} - \frac{1}{8}}}{a^{\frac{4}{9} - \frac{1}{2}}} $}[/tex]
[tex]\Large \text{$ \sf \dfrac{\dfrac{\dfrac{\sqrt[\sf 4]{\sf a}}{\sqrt[\sf 6]{\sf a}}}{\sqrt[\sf 8]{\sf a}}}{\dfrac{\sqrt[\sf 3]{\sf a}\:.\:\sqrt[\sf 9]{\sf a}}{\sqrt{\sf a}}} = \dfrac{a^{-\frac{1}{24} }}{a^{-\frac{1}{18}}} $}[/tex]
[tex]\Large \text{$ \sf \dfrac{\dfrac{\dfrac{\sqrt[\sf 4]{\sf a}}{\sqrt[\sf 6]{\sf a}}}{\sqrt[\sf 8]{\sf a}}}{\dfrac{\sqrt[\sf 3]{\sf a}\:.\:\sqrt[\sf 9]{\sf a}}{\sqrt{\sf a}}} = a^{-\frac{1}{24} + \frac{1}{18}} $}[/tex]
[tex]\Large \text{$ \sf \dfrac{\dfrac{\dfrac{\sqrt[\sf 4]{\sf a}}{\sqrt[\sf 6]{\sf a}}}{\sqrt[\sf 8]{\sf a}}}{\dfrac{\sqrt[\sf 3]{\sf a}\:.\:\sqrt[\sf 9]{\sf a}}{\sqrt{\sf a}}} = a^{\frac{1}{72}} $}[/tex]
[tex]\Large \boxed{\boxed{\text{$ \sf \dfrac{\dfrac{\dfrac{\sqrt[\sf 4]{\sf a}}{\sqrt[\sf 6]{\sf a}}}{\sqrt[\sf 8]{\sf a}}}{\dfrac{\sqrt[\sf 3]{\sf a}\:.\:\sqrt[\sf 9]{\sf a}}{\sqrt{\sf a}}} = \sqrt[\sf 72]{\sf a} $}}}[/tex]