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Question Number 163408 by mathlove last updated on 06/Jan/22

$$\sqrt[3]{1+\frac{2}{9}\sqrt{21}}+\sqrt[3]{1-\frac{2}{9}\sqrt{21}}=?$$

Answered by mr W last updated on 06/Jan/22

$$\sqrt[3]{1+\frac{2}{9}\sqrt{21}}+\sqrt[3]{1-\frac{2}{9}\sqrt{21}}=x$$$$1+\frac{2}{9}\sqrt{21}+1-\frac{2}{9}\sqrt{21}+3x\sqrt[3]{(1+\frac{2}{9}\sqrt{21})(1-\frac{2}{9}\sqrt{21}})=x^3$$$$x^3+x-2=0$$$$(x-1)(x^2+x+2)=0$$$$\Rightarrow x=1$$

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