<s>[INST] Determine (without using a calculator) which of the following is bigger: $1+\sqrt[3]{2}$ or $\sqrt[3]{12}$ I have encountered the following question in a highschool book in the subject of powers. and, it seems I can't solve it.... 
Determine (without using a calculator) which of the following is bigger: $1+\sqrt[3]{2}$ or $\sqrt[3]{12}$
Any ideas?
Thank you!
Shir
 [/INST] After reading your helpfull answers, got something of my own:
$ (1+\sqrt[3]{2})<\sqrt[3]{12} \iff (1+\sqrt[3]{2})^3<12 \iff 1+3\sqrt[3]{2}+3\sqrt[3]{4} +2 < 12 \iff 3\sqrt[3]{2}+3\sqrt[3]{4} < 9 \iff \sqrt[3]{2}+\sqrt[3]{4} < 3 \iff 2+3\sqrt[3]{2}+3\sqrt[3]{4}+4 < 27 \iff 3\sqrt[3]{2}+3\sqrt[3]{4} < 21 \iff \sqrt[3]{2} + \sqrt[3]{4} < 7 $ and the last inequality is true since $\sqrt[3]{2}<2$ and $\sqrt[3]{4}<4$ 
Thank you all for your answers!!
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