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Question Number 99094 by bemath last updated on 18/Jun/20

$$\mathrm{find}\sqrt[3]{9+9\sqrt[3]{9+9\sqrt[3]{9+9\sqrt[3]{9+...}}}}-$$$$\sqrt{8-\sqrt{8-\sqrt{8+\sqrt{8-\sqrt{8-\sqrt{8-\sqrt{8-\sqrt{}}}}...}}}}$$

Answered by bobhans last updated on 18/Jun/20

$$\mathrm{y}=\sqrt[3]{9+9\mathrm{y}}\Rightarrow {\mathrm{y}}^3-9\mathrm{y}-9=0$$$$\mathrm{x}=\sqrt{8-\sqrt{8-\sqrt{8+\mathrm{x}}}}\Rightarrow {\mathrm{x}}^3+{3\mathrm{x}}^2-6\mathrm{x}-17=0$$$$\Rightarrow {(\mathrm{x}+1)}^3-9(\mathrm{x}+1)-9=0$$$$\mathrm{let}\mathrm{y}=\mathrm{x}+1\Rightarrow {(\mathrm{x}+1)}^3-9(\mathrm{x}+1)-9$$$$\mathrm{then}{(\mathrm{x}+1)}^3-9(\mathrm{x}+1)-9-({(\mathrm{x}+1)}^3-9(\mathrm{x}+1)-9)=0$$

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