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Question Number 99089 by bramlex last updated on 18/Jun/20

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

Answered by MJS last updated on 18/Jun/20

$$a=\sqrt[3]{9+\sqrt[3]{9+...}}$$$$a^3=9+a$$$$a^3-a-9=0$$$$a=\sqrt[3]{\frac{9}{2}+\frac{\sqrt{6549}}{18}}+\sqrt[3]{\frac{9}{2}-\frac{\sqrt{6549}}{18}}$$$$b=\sqrt{8-\sqrt{8-...}}$$$$b^2=8-b$$$$b=-\frac{1}{2}+\frac{\sqrt{33}}{2}$$$$a-b=\sqrt[3]{\frac{9}{2}+\frac{\sqrt{6549}}{18}}+\sqrt[3]{\frac{9}{2}-\frac{\sqrt{6549}}{18}}+\frac{1-\sqrt{33}}{2}$$

Commented by bemath last updated on 18/Jun/20

$$\mathrm{cardano}$$

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