The following equation is true. How is the left simplified to match the right? This has been bugging me for hours now.

Question

anonymous · Answer

$$\sqrt[3]{7+5 \sqrt{2} } = 1 + \sqrt{2}$$

anonymous · Answer

Try cubing both sides.

anonymous · Answer

$$\large 7+5\sqrt{2}=(1+\sqrt{2})^3$$

anonymous · Answer

Sorry I wasn't clear on this: Considering $$\sqrt[3] {7+5 \sqrt {2}}$$ by it self. What are the steps necessary to simplify the equation into a rational number.  in this case I cannot cube both sides.

zugzwang · Answer

It does help to cube both sides though :)
Makes working backwards much clearer

anonymous · Answer

Ah, verifying an identity, yes that does make it more difficult.

zugzwang · Answer

To be clear, cubing both sides is only a sketch of it...
Perhaps it would be simpler if you write
$$7 + 5\sqrt{2}$$
as
$$1 + 3\sqrt{2} + 6 + 2\sqrt{2}$$

zugzwang · Answer

Such that
$$\sqrt[3]{7+5\sqrt{2}}=\sqrt[3]{1 + 3\sqrt{2}+6+2\sqrt{2}}$$
$$=\sqrt[3]{\left( 1 + \sqrt{2} \right)^{3}}$$

anonymous · Answer

That defiantly answered my question on how to try thinking when doing so. Didn't think of the interior by it self as a sort of a quadratic equation.

zugzwang · Answer

A little creativity and clever manipulation (?) could help a whole lot :D