Simplifying Cube Roots Containing a Square Root

Question

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.

Answer 1

You're almost there. Note that $2=\sqrt{2}\sqrt{2}$, then agrue as follows $$x=20\sqrt[3]{2\sqrt{2}}=20\sqrt[3]{(\sqrt{2}\sqrt{2})\sqrt{2}}=20\sqrt[3]{\sqrt{2}^3}=20\sqrt{2}$$

Answer 2

$$16000\sqrt2=1000\cdot2^4\cdot2^{1/2}=10^3\cdot2^{4+1/2}$$

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

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

Answer 3

$$16000\sqrt 2=8000\times 2\sqrt 2=(20)^3\cdot (\sqrt 2)^3=(20\sqrt 2)^3$$