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Is there a method to proving whether nested roots like $\sqrt{4 + 2\sqrt{3}} - \sqrt{3}$ are rational or not?

For simpler radicals, I can try using a contradiction, assume that they do equal some rational number and use algebra to simplify and show how one side of the equation is clearly rational but the other reduced to some well known root, like 2 or 3, and conclude irrationality.

lulu
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armorlord
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1 Answers1

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$$\sqrt{(4+2\sqrt3)}-\sqrt3=\\ \sqrt{(1+3+2\sqrt3)}-\sqrt3=\\ \sqrt{(1+\sqrt3^2+2\sqrt3)}-\sqrt3=\\ \sqrt{(1+\sqrt3)^2}-\sqrt3=\\ 1+\sqrt3-\sqrt3=1$$

Khosrotash
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