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\sqrt[3]{18+\sqrt{84-\sqrt{4+5}}}
Calculate the square root of 25 and get 5.
\sqrt[3]{18+\sqrt{84-\sqrt{9}}}
Add 4 and 5 to get 9.
\sqrt[3]{18+\sqrt{84-3}}
Calculate the square root of 9 and get 3.
\sqrt[3]{18+\sqrt{81}}
Subtract 3 from 84 to get 81.
\sqrt[3]{18+9}
Calculate the square root of 81 and get 9.
\sqrt[3]{27}
Add 18 and 9 to get 27.
3
Calculate \sqrt[3]{27} and get 3.