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\sqrt[3]{333+\sqrt[3]{987+13}}
Calculate \sqrt[3]{2197} and get 13.
\sqrt[3]{333+\sqrt[3]{1000}}
Add 987 and 13 to get 1000.
\sqrt[3]{333+10}
Calculate \sqrt[3]{1000} and get 10.
\sqrt[3]{343}
Add 333 and 10 to get 343.
7
Calculate \sqrt[3]{343} and get 7.