How can you proof that the sum of three roots is irrational?

Question

I would like to know how to prove that $\sqrt{2} + \sqrt{5} + \sqrt{7}$ is an irrational number. I know how to do the proof for a sum of two roots. Can I just define $\sqrt{2} + \sqrt{5} :=c$ and then prove that $c$ and $c + \sqrt{7}$ are both irrational, when I treat $c$ as a single number? If this way is kind of wrong or won't work, is there another way to do the proof?

Answer 1

One way to prove that the sum of two square roots (e.g. $\sqrt 2 + \sqrt 5$) is irrational is to suppose not, and square once. Upon rearrangement, you'll find a contradiction. Here we can do the same thing, squaring twice with some algebra: Suppose $$r = \sqrt 2 + \sqrt 5 + \sqrt7$$ were rational; then

$$r - \sqrt7 = \sqrt2 + \sqrt5$$

Squaring both sides, you'll find that

$$r^2 + 7 - 2 r \sqrt7 = 2 + 5 + 2\sqrt{10}$$

Simplify, place both radicals one one side of the equation, and square again; try to get a contradiction from here.

Of course, this doesn't always work: There are triplets of numbers such that the corresponding $r$ is rational, so carefully identify where the contradiction is.

Answer 2

(A stronger result for reference)

If $a_i$ are postive integers, then $\sum \sqrt[n]{a_i}$ is rational iff $\sqrt[n]{a_i}$ is.

Answer 3

You might want to take a look at www.thehcmr.org/issue2_1/mfp.pdf.