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Question Number 144760 by mathdanisur last updated on 28/Jun/21

$$\sqrt[3]{1+\sqrt{x}}+\sqrt[3]{1-\sqrt{x}}=\sqrt[3]{5}$$$$Find\mathit{x}=?$$

Answered by Olaf_Thorendsen last updated on 28/Jun/21

$$\mathrm{Let}\alpha =\sqrt[3]{1+\sqrt{x}}\mathrm{and}\beta =\sqrt[3]{1-\sqrt{x}}$$$$\alpha +\beta =\sqrt[3]{5}$$$${(\alpha +\beta )}^3={\alpha }^3+{\beta }^3+3\alpha \beta (\alpha +\beta )=5$$$$(1+\sqrt{x})+(1-\sqrt{x})+3\sqrt[3]{5}\sqrt[3]{1-x}=5$$$$\sqrt[3]{1-x}=\frac{1}{\sqrt[3]{5}}$$$$1-x=\frac{1}{5}$$$$x=\frac{4}{5}$$

Commented by mathdanisur last updated on 29/Jun/21

$$coolSirthankyou$$

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