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\sqrt{2+\sqrt[2]{3+1}}
Calculate \sqrt[3]{1} and get 1.
\sqrt{2+\sqrt[2]{4}}
Add 3 and 1 to get 4.
\sqrt{2+2}
Calculate \sqrt[2]{4} and get 2.
\sqrt{4}
Add 2 and 2 to get 4.
2
Calculate the square root of 4 and get 2.