# Can we find using rule and

coud we find using rule and compass :Confused: an exact segment of value:

$$\sqrt (n)$$ for every natural number n

- the same but for $$n^{1/m}$$ where n and m are positive integers

- given a segment of length a known can we find an exact segment of length $$a 2^{-1/2}$$

for the case m=2 using Pythagorean theorem is easy to find but what about the other cases.

It is quite possible to take the square root of a number, and use the four arithmetic functions. However, everything by ruler and compass is equivalent to solving linear and quadratic equations.

In general we can not find for integers, N &M, N^(1/M).

Last edited:
HallsofIvy
$\sqrt{n}$ satisfies the equation x2- n= 0 and so is either algebraic of order 2 (if x2- n is irreducible- cannot be factored with integer coefficients) or algebraic of order 1: both of which are powers of two.
We can construct a segment of length $n^{1/m}$ if and only if m is a power of two.