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Question Number 212531 by RojaTaniya last updated on 16/Oct/24

Answered by a.lgnaoui last updated on 17/Oct/24

$$\mathbf{x}=1{+}^7\sqrt{2}+{\left({}^7\sqrt{2}\right)}^2+{\left({}^7\sqrt{2}\right)}^3+{\left({}^7\sqrt{2}\right)}^4+{\left({}^7\sqrt{2}\right)}^5+{\left(\sqrt{2}\right)}^6$$$$=\frac{1-{\left({}^7\sqrt{2}\right)}^7}{1-\left({}^7\sqrt{2}\right)}=\frac{1}{{}^7\sqrt{2}-1}\Rightarrow \frac{1}{\mathbf{x}}{=}^7\sqrt{2}-1$$$$$$$$\frac{1}{{\mathbf{x}}^3}-\frac{3}{\mathbf{x}}=\frac{1}{\mathbf{x}}(\frac{1}{{\mathbf{x}}^2}-3)=({}^7\sqrt{2}-1){[\left({}^7\sqrt{2}\right)-1)}^2-3]$$$$$$$$\frac{1}{{\mathbf{x}}^3}-\frac{3}{\mathbf{x}}={\left({}^7\sqrt{2}\right)}^3-3{\left({}^7\sqrt{2}\right)}^2+2$$$$$$

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