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\sqrt[3]{55+\sqrt{78+3}}
Calculate \sqrt[3]{27} and get 3.
\sqrt[3]{55+\sqrt{81}}
Add 78 and 3 to get 81.
\sqrt[3]{55+9}
Calculate the square root of 81 and get 9.
\sqrt[3]{64}
Add 55 and 9 to get 64.
4
Calculate \sqrt[3]{64} and get 4.