Help Me please

$\quad\sqrt[3]{16}\,\times\,7\sqrt[3]{2} \\ =\\ \quad \sqrt[3]{16}\,\times\,7\,\times\,\sqrt[3]{2} \\ =\\ \quad \sqrt[3]{16}\,\times\,\sqrt[3]{2}\,\times\,7 \\ =\\ \quad \sqrt[3]{16\,\times\,2}\,\times\,7 \\ =\\ \quad \sqrt[3]{2\,\times\,2\,\times\,2\,\times2\,\times\,2}\,\times\,7 \\ =\\ \quad \sqrt[3]{2^3\,\times\,2^2}\,\times\,7 \\ =\\ \quad \sqrt[3]{2^3}\,\times\,\sqrt[3]{2^2}\,\times\,7 \\ =\\ \quad 2\,\times\,\sqrt[3]{4}\,\times\,7 \\ =\\ \quad 2\,\times\,7\,\times\,\sqrt[3]{4} \\ =\\ \quad 14\,\times\,\sqrt[3]{4} \\ =\\ \quad 14\sqrt[3]{4} \\ $

Thanks Hectitar but I think the correct answer is

686 x 2^5/6

I checked it with Wolfram Alpha

(http://www.wolframalpha.com/input/?i=%5E3%E2%88%9A16+x+7%5E3%E2%88%9A2)

Plus I checked it with another calculator and the answer was: 686 x 2^5/6

Just a  slight mis-interpretation......no big deal

So we have

³√16 * 7³ √2   =

³√16  * 343√2  =

³√8  * ³√2  *  343√2  =

2 * ³√2  *  343√2  =

2 * 343  *  (2)^1/3  * (2)^1/2   =

686 * (2)^2/6 * (2)^3/6   =

686 * (2)^5/6  

Thank You All!!