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# Rational Roots

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions

In this problem you are given that $a$, $b$ and $c$ are natural numbers. You have to show that if $\sqrt{a}+\sqrt{b}$ is rational then it is a natural number.

You could use the fact that if $\sqrt{a}+\sqrt{b}$ is rational then so is its square which means that $\sqrt ab $ is also rational. Knowing this the next step is to use $$\sqrt{a}(\sqrt{a}+\sqrt{b}) = a+\sqrt{ab}$$ to show that $\sqrt a$ is rational and to do likewise for $b$.

This is all you need because it has been proved that if $\sqrt a$ is rational then $a$ must be a square number.

Try to apply this method and then to extend it to three variables for the last part.

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.