For this case we must find an expression equivalent to:
[tex]\frac{4\sqrt{6}}{\sqrt[3]{2}}[/tex]
We multiply by:
[tex](\frac{(\sqrt[3]{2})^2}{(\sqrt[3]{2})^2})\\\frac{4\sqrt{6}}{\sqrt[3]{2}}*(\frac{(\sqrt[3]{2})^2}{(\sqrt[3]{2})^2})=[/tex]
By definition of multiplication of powers of the same base we have:
[tex]a^n*a^m=a^{m+n}\\\frac{4\sqrt{6}*(\sqrt[3]{2})^2}{(\sqrt[3]{2})^3}=\\\frac{4\sqrt{6}*(\sqrt[3]{2})^2}{2}=[/tex]
Move the exponent within the radical:
[tex]\frac{4\sqrt{6}*\sqrt[3]{2^2}}{2}=\\\frac{4\sqrt{6}*\sqrt[3]{4}}{2}=[/tex]
We rewrite:
[tex]4^{\frac{1}{3}}=4^{\frac{2}{6}}=\sqrt[6]{4^2}\\6^{\frac{1}{2}}=6^{\frac{3}{6}}=\sqrt[6]{6^3}[/tex]
Rewriting the expression:
[tex]\frac{4\sqrt[6]{4^2}\sqrt[6]{6^3}}{2}=\\\frac{4\sqrt[6]{16*216}}{2}=\\\frac{4\sqrt[6]{3456}}{2}=\\\frac{4\sqrt[6]{2^6*54}}{2}=\\\frac{8\sqrt[6]{54}}{2}=\\4\sqrt[6]{54}[/tex]
Answer:
[tex]4\sqrt[6]{54}[/tex]