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Question Number 123576 by bramlexs22 last updated on 26/Nov/20

Commented by talminator2856791 last updated on 26/Nov/20

$$\mathrm{the}\mathrm{image}\mathrm{is}\mathrm{not}\mathrm{so}\mathrm{clear}$$

Commented by benjo_mathlover last updated on 26/Nov/20

$$\sqrt[3]{\sqrt[3]{2}-1}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$$$$findthevalueofa+b+c$$

Answered by Dwaipayan Shikari last updated on 26/Nov/20

$$\sqrt[3]{\sqrt[3]{2}-1}=\sqrt[3]{p}+\sqrt[3]{c}$$$$\sqrt[3]{2}-1=p+c+3\sqrt[3]{pc}(\sqrt[3]{p}+\sqrt[3]{c})$$$$\sqrt[3]{2}-1=p+c+3\sqrt[3]{pc}\left(\sqrt[3]{\sqrt[3]{2}-1}\right)$$$$\sqrt[3]{2}=3\sqrt[3]{pc}\left(\sqrt[3]{\sqrt[3]{2}-1}\right)$$$$2=27pc(\sqrt[3]{2}-1)$$$$p+c=-1\Rightarrow p+\frac{2}{(\sqrt[3]{2}-1)27p}=-1$$$$p^2+p+\frac{2}{(\sqrt[3]{2}-1)27}=0$$$$p=\frac{-1\pm \sqrt{1-\frac{4}{(\sqrt[3]{2}-1)27}}}{2}$$$$p=-\frac{1}{2}\pm \frac{1}{2}\sqrt{1-\frac{4}{(\sqrt[3]{2}-1)27}}$$$$c=-\frac{1}{2}\pm \frac{1}{2}\sqrt{1-\frac{4}{(\sqrt[3]{2}-1)27}}$$$$p=(a+b+3\sqrt[3]{ab}(\sqrt[3]{a}+\sqrt[3]{b}))$$$$$$$$$$

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