simplify $c=\sqrt{290-143\sqrt2}$

Question

I am trying to simplify $c=\sqrt{290-143\sqrt2}.$ I am solving a triangle and I got that $c^2=290-143\sqrt2.$ I have tried to use the formula for $\sqrt{a\pm\sqrt{b}}$ but it seemed useless at the end. Can you give me a hint? I want to remove the square root.

Answer 1

It is impossible to remove the square root and get something of the form $a+b\sqrt c$ where $a,b,c$ are rational numbers. If that were possible, the minimal polynomial of the given number would be quadratic. But that of $\sqrt{290-143\sqrt2}$ is a quartic $x^4-580x^2+43202=0$.

Answer 2

Hint

$$(\sqrt{a}-\sqrt{b})^2=290-143\sqrt{2}$$ $$a+b-2\sqrt{ab}=290-143\sqrt{2}$$ so, $$a+b=290 \quad \text{and} \quad 4ab=2.143^2$$ which is impossible to solve for $a$ and $b$ rational numbers.