Question

simplify :
$( \frac{12 {}^{ \frac{1}{6} } }{27 \frac{1}{6} } ) {}^{ \frac{6}{8} }$
​

Answer 1

$\Large\green\dag\:\:$ $\underline{\boxed{\Large{\tt{\red{Solution}}}}}$ $\:\:\:\Large\green\dag$

$\pink\longrightarrow\:\:\:\tt\purple{\bigg( \dfrac{12 {}^{ \frac{1}{6} } }{27 \frac{1}{6} } \bigg) {}^{ \frac{6}{8} }}$

$\pink\longrightarrow\:\:\:\tt\blue{\bigg( \dfrac{12 {}^{ \frac{1}{6} } }{ \frac{ 163}{6} } \bigg) {}^{ \frac{3}{4} }}$

$\pink\longrightarrow\:\:\:\tt\orange{\bigg( \dfrac{6 \times 12 {}^{ \frac{1}{6} } }{163} \bigg) {}^{ \frac{3}{4} }}$

$\pink\longrightarrow\:\:\:\tt\green{\bigg( \dfrac{6 \sqrt[6]{12} }{163}\bigg) {}^{ \frac{3}{4} }}$

$\pink\longrightarrow\:\:\:\tt\purple{ \sqrt[4]{\bigg( \dfrac{6 \sqrt[6]{12} }{163} \bigg) {}^{3}} }$

$\pink\longrightarrow\:\:\:\tt\blue{ \dfrac{ \sqrt[4]{432 \sqrt{3} } }{ \sqrt[4]{163} {}^{ {}^{3} } }}$

$\pink\longrightarrow\:\:\:\tt\orange{ \dfrac{2 \sqrt[4]{27 \sqrt{3} } }{ \sqrt[4]{163} {}^{ {}^{3} } }}$

$\pink \longrightarrow\:\:\:\tt\green{\dfrac{2 \sqrt[4]{ \sqrt{3} {}^{7} } }{ \sqrt[4]{163} {}^{2} } }$

$\pink\longrightarrow\:\:\:\tt\purple{ \dfrac{2 \sqrt[8]{3 {}^{7} } \: \: \sqrt[8]{163 {}^{2} } }{163}}$

$\pink\longrightarrow\:\:\:\tt\blue{ \dfrac{2 \sqrt[8 ]{3 {}^{7} \times {163}^{2} } }{163}}$

$\pink \longrightarrow\:\:\:\tt\orange{\dfrac{2 \sqrt[8]{2187 \times 26569} }{163} }$

$\pink\longrightarrow\:\:\:\tt\green{ \dfrac{2 \sqrt[8]{58106403} }{163} }$

$\tt\red{In\:\: Alternative \:\:Form \:\::}$

$\underline{\boxed{\tt{\orange{\dfrac{2 \sqrt[8]{58106403} }{163}\:\approx\:0.114649}}}}$