Making the segment with given length $\sqrt[3]{2}$?

Question

Using Pythagoras' Theorem we can make the segment with given length of Square root of natural numbers. For example the segment of given length
The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1.

Now how we can make the segment with given length $\sqrt[3]2$, the third root of $2$?

Thank you

Answer 1

It follows from Wantzel's theorem that the cube root of $2$ cannot be constructed by ruler and compass.

You can construct it with other tools; a method that goes back to Menaechmus is by intersecting a parabola with a hyperbola.

With analytic geometry it's easier: consider the curves of equations $xy=2$ and $y=x^2$. Then the abscissa of the intersection point satisfies $x^3=2$ and so is the solution to your problem.