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I was doing a problem today, and arrived at the (correct) answer of $x^3 = 16000\sqrt2$

Obviously I want to simplify this further. My text book jumps straight to $x = 20\sqrt2$ with no explanation.

In attempting to simplify it, I've got:

$x = \sqrt[3] {16000\sqrt{2}}$

$x = (\sqrt[3]{16000})(\sqrt[3]{\sqrt2})$

$x = (\sqrt[3]{800})(\sqrt[3]{2})(\sqrt[3]{\sqrt2})$

$x = 20\sqrt[3]{2\sqrt{2}}$

Could someone please explain the steps needed to get to $20\sqrt{2}$?

I accept this is trivial, but I'm stumped. Thanks in advance.

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    $\begingroup$ Note that $2=\sqrt{2}\sqrt{2}$ $\endgroup$ – John Joy Aug 28 '15 at 12:30
  • $\begingroup$ @JohnJoy Was there some reason you did not want to post that as an answer? It seems to be the most applicable response to this question as asked. (I would have upvoted it.) $\endgroup$ – David K Aug 28 '15 at 13:52
  • $\begingroup$ @David K I took a second look at the other answers and realized that mine was just a restatement of mathlove's answer. Feel free to upvote my comment though. $\endgroup$ – John Joy Aug 28 '15 at 15:32
  • $\begingroup$ @JohnJoy I was thinking specifically of how one gets "unstuck" after writing $\sqrt[3]{2\sqrt{2}}$. But you're right, mathlove's answer also gives a big clue about this and OP accepted it, so I guess that's enough. $\endgroup$ – David K Aug 28 '15 at 17:11
  • $\begingroup$ Please feel free to elaborate further - I've enjoyed your comments so far. $\endgroup$ – Bangkockney Aug 28 '15 at 17:21
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You're almost there. Note that $2=\sqrt{2}\sqrt{2}$, then agrue as follows $$x=20\sqrt[3]{2\sqrt{2}}=20\sqrt[3]{(\sqrt{2}\sqrt{2})\sqrt{2}}=20\sqrt[3]{\sqrt{2}^3}=20\sqrt{2}$$

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  • $\begingroup$ This is a great answer too. I particularly appreciate how you followed on from my working. Thanks for this. $\endgroup$ – Bangkockney Aug 29 '15 at 1:18
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$$16000\sqrt 2=8000\times 2\sqrt 2=(20)^3\cdot (\sqrt 2)^3=(20\sqrt 2)^3$$

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  • $\begingroup$ Thank you. This is wonderfully clear to me. I seem to regularly forget to think in this way. $\endgroup$ – Bangkockney Aug 28 '15 at 10:34
  • $\begingroup$ @Bangkockney: You are welcome! $\endgroup$ – mathlove Aug 28 '15 at 10:36
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$$16000\sqrt2=1000\cdot2^4\cdot2^{1/2}=10^3\cdot2^{4+1/2}$$

$\implies$ the principal value of $$\sqrt[3]{6000\sqrt2}=10\cdot2^{3/2}$$

Now $2^{3/2}=2^{1+1/2}=2\sqrt2$

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