Question

01) Determine o valor da expressão abaixo:
$\huge\mathtt{\frac{\sqrt[3]{\frac{1}{2}\sqrt[4]{\frac{3}{4}\sqrt[5]{\frac{5}{6}}}}\times\sqrt[3]{\frac{3}{2}\sqrt[4]{\frac{5}{3}\sqrt[5]{\frac{4}{5}}}}}{\sqrt[3]{\frac{3}{4}\sqrt[4]{\frac{5}{4}\sqrt[5]{\frac{2}{3}}}}}}$

Answer 1

Resposta:

1

Explicação passo a passo:

$1)\sqrt[3]{\frac{1}{2}\sqrt[4]{\frac{3}{4} \sqrt[5]{\frac{5}{6} } } }= \sqrt[3]{\frac{1}{2}\sqrt[4]{\frac{3}{4} (\frac{5}{6} )^{\frac{1}{5} } } }\\\sqrt[3]{\frac{1}{2}\sqrt[4]{\frac{3}{4} \sqrt[5]{\frac{5}{6} } } }=\sqrt[3]{\frac{1}{2}[\frac{3}{4} (\frac{5}{6} )^{\frac{1}{5} } ]^{\frac{1}{4} } }\\\sqrt[3]{\frac{1}{2}\sqrt[4]{\frac{3}{4} \sqrt[5]{\frac{5}{6} } } }=\sqrt[3]{\frac{1}{2}(\frac{3}{4})^{\frac{1}{4} } (\frac{5}{6} )^{\frac{1}{20} } }$

$\sqrt[3]{\frac{1}{2}\sqrt[4]{\frac{3}{4} \sqrt[5]{\frac{5}{6} } } }=[(\frac{1}{2} )(\frac{3}{4} )^{\frac{1}{4} }(\frac{5}{6} )^{\frac{1}{20} }]^{\frac{1}{3} }\\\sqrt[3]{\frac{1}{2}\sqrt[4]{\frac{3}{4} \sqrt[5]{\frac{5}{6} } } }=(\frac{1}{2} )^{\frac{1}{3} }(\frac{3}{4} )^{\frac{1}{12} }(\frac{5}{6} )^{\frac{1}{60} }$

$\sqrt[3]{\frac{1}{2}\sqrt[4]{\frac{3}{4} \sqrt[5]{\frac{5}{6} } } }=(2^{-\frac{1}{3} })(3^{\frac{1}{12} })(4^{-\frac{1}{12} })(5^{\frac{1}{60} })(6^{-\frac{1}{60} })$

$2)\sqrt[3]{\frac{3}{2}\sqrt[4]{\frac{5}{3} \sqrt[5]{\frac{4}{5} } } }=\sqrt[3]{\frac{3}{2}\sqrt[4]{\frac{5}{3}(\frac{4}{5} )^{\frac{1}{5} } } } \\\sqrt[3]{\frac{3}{2}\sqrt[4]{\frac{5}{3} \sqrt[5]{\frac{4}{5} } } }=\sqrt[3]{\frac{3}{2}[\frac{5}{3}(\frac{4}{5} )^{\frac{1}{5} } ]^{\frac{1}{4} } }\\\sqrt[3]{\frac{3}{2}\sqrt[4]{\frac{5}{3} \sqrt[5]{\frac{4}{5} } } }=\sqrt[3]{\frac{3}{2}(\frac{5}{3} )^{\frac{1}{4} }(\frac{4}{5} )^{\frac{1}{20} } }$

$\sqrt[3]{\frac{3}{2}\sqrt[4]{\frac{5}{3} \sqrt[5]{\frac{4}{5} } } }=[(\frac{3}{2} )(\frac{5}{3} )^{\frac{1}{4} }(\frac{4}{5} )^{\frac{1}{20} }]^{\frac{1}{3} }\\\sqrt[3]{\frac{3}{2}\sqrt[4]{\frac{5}{3} \sqrt[5]{\frac{4}{5} } } }=(\frac{3}{2} )^{\frac{1}{3} }(\frac{5}{3} )^{\frac{1}{12} }(\frac{4}{5} )^{\frac{1}{60} }$

$\sqrt[3]{\frac{3}{2}\sqrt[4]{\frac{5}{3} \sqrt[5]{\frac{4}{5} } } }=(2^{-\frac{1}{3} })(3^{\frac{1}{4} })(4^{\frac{1}{60} })(5^{\frac{1}{15} })$

$3)Numerador:\ (2^{-\frac{2}{3} })(3^{\frac{1}{3} })(4^{-\frac{1}{15} })(5^{\frac{1}{12} })(6^{-\frac{1}{60} })$

$4)Denominador:\\\sqrt[3]{\frac{3}{4}\sqrt[4]{\frac{5}{4}\sqrt[5]{\frac{2}{3} } } }= \sqrt[3]{\frac{3}{4}\sqrt[4]{\frac{5}{4} (\frac{2}{3} )^{\frac{1}{5} }} }\\\sqrt[3]{\frac{3}{4}\sqrt[4]{\frac{5}{4}\sqrt[5]{\frac{2}{3} } } }= \sqrt[3]{\frac{3}{4}[(\frac{5}{4} )(\frac{2}{3} )^{\frac{1}{5} }]^{\frac{1}{4} } } \\\sqrt[3]{\frac{3}{4}\sqrt[4]{\frac{5}{4}\sqrt[5]{\frac{2}{3} } } }= \sqrt[3]{(\frac{3}{4} )(\frac{5}{4} )^{\frac{1}{4} }(\frac{2}{3} )^{\frac{1}{20} }}$

$\sqrt[3]{\frac{3}{4}\sqrt[4]{\frac{5}{4}\sqrt[5]{\frac{2}{3} } } }= [(\frac{3}{4} )(\frac{5}{4} )^{\frac{1}{4} }(\frac{2}{3} )^{\frac{1}{20} }]^{\frac{1}{3} }\\\sqrt[3]{\frac{3}{4}\sqrt[4]{\frac{5}{4}\sqrt[5]{\frac{2}{3} } } }= (\frac{3}{4} )^{\frac{1}{3} }(\frac{5}{4} )^{\frac{1}{12} }(\frac{2}{3} )^{\frac{1}{60} }\\\sqrt[3]{\frac{3}{4}\sqrt[4]{\frac{5}{4}\sqrt[5]{\frac{2}{3} } } }= (2^{\frac{1}{60} })(3^{\frac{19}{60} })(4^{-\frac{5}{12} })(5^{\frac{1}{12} })$

$5)Resultado:\\(2^{\frac{-41}{60} })(3^{\frac{1}{60} })(4^{\frac{21}{60} })(6^{-\frac{1}{60} })\\(2^{\frac{-41}{60} })(3^{\frac{1}{60} })[(2^{2})^{\frac{21}{60} }][(2*3)^{-\frac{1}{60} }]\\(2^{\frac{-41}{60} })(3^{\frac{1}{60} })(2^{\frac{42}{60} })(2^{\frac{-1}{60} })(3^{\frac{-1}{60} })\\2^{0}*3^{0}\\1$