# Exercise ID278

Algebra → Roots → Simplifying
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Simplify the following expression:

$\sqrt{\frac{\left(1+a\right)·\sqrt[3]{1+a}}{3a}}·\sqrt[3]{\frac{\sqrt{3}}{9+18{a}^{-1}+9{a}^{-2}}}$

$\sqrt{\frac{\left(1+a\right)·\sqrt[3]{1+a}}{3a}}·\sqrt[3]{\frac{\sqrt{3}}{9+18{a}^{-1}+9{a}^{-2}}}$ $=\sqrt{\frac{\sqrt[3]{\left(1+a\right)·{\left(1+a\right)}^{3}}}{3a}}·\sqrt[3]{\frac{\sqrt{3}}{9+\frac{18}{a}+\frac{9}{{a}^{2}}}}$ $=\frac{\sqrt[6]{{\left(1+a\right)}^{4}}}{\sqrt{3a}}·\sqrt[3]{\frac{\sqrt{3}}{\frac{9{a}^{2}+18a+9}{{a}^{2}}}}$ $=\frac{\sqrt[6]{{\left(1+a\right)}^{4}}}{\sqrt{3a}}·\sqrt[3]{\frac{\sqrt{3}{a}^{2}}{9{a}^{2}+18a+9}}$ $=\frac{\sqrt[6]{{\left(1+a\right)}^{4}}}{\sqrt{3a}}·\sqrt[3]{\frac{\sqrt{3{a}^{4}}}{9\left({a}^{2}+2a+1\right)}}$ $=\sqrt[6]{\frac{{\left(1+a\right)}^{4}}{27{a}^{3}}}·\sqrt[3]{\frac{\sqrt{3{a}^{4}}}{\sqrt{81{\left({a}^{2}+2a+1\right)}^{2}}}}$ $=\sqrt[6]{\frac{{\left(1+a\right)}^{4}}{27{a}^{3}}}·\sqrt[6]{\frac{3{a}^{4}}{81{\left({a}^{2}+2a+1\right)}^{2}}}$ $=\sqrt[6]{\frac{{\left(1+a\right)}^{4}}{27{a}^{3}}·\frac{3{a}^{4}}{81{\left({\left(1+a\right)}^{2}\right)}^{2}}}$ $=\sqrt[6]{\frac{{\left(1+a\right)}^{4}}{27{{a}}^{3}}·\frac{3{a}^{{4}}}{81{\left(1+a\right)}^{4}}}$ $=\sqrt[6]{\frac{{3}a}{{27}·81}}$ $=\sqrt[6]{\frac{a}{9·81}}$ $=\frac{\sqrt[6]{a}}{\sqrt[6]{9·81}}$ $=\frac{\sqrt[6]{a}}{\sqrt[6]{729}}$ $=\frac{\sqrt[6]{a}}{3}$
$A=\sqrt{\frac{\left(1+a\right)·\sqrt[3]{1+a}}{3a}}·\sqrt[3]{\frac{\sqrt{3}}{9+18{a}^{-1}+9{a}^{-2}}}=\frac{\sqrt[6]{a}}{3}$