How to find if the result of a problem is a rational or irrational number without solving the problem?

Question

I have got a set of problems

√2-√3
√8/√2

And i have to find if the result of the problem is going to be a rational or irrational number.How can i do that considering that we know which part of problem is a rational and which part is irrational number?

Answer 1

Assume that $\sqrt2-\sqrt3$ is rational.

Then, $\sqrt2+\sqrt3 = \dfrac1{\sqrt3-\sqrt2}$ is also rational.

Therefore, $(\sqrt2+\sqrt3)+(\sqrt2-\sqrt3)=2\sqrt2$ is also rational.

Therefore, $\sqrt2$ is rational.

This is a contradiction, so $\sqrt2-\sqrt3$ is irrational.

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$\dfrac{\sqrt8}{\sqrt2}=2$ is rational.