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Question Number 176598 by Rasheed.Sindhi last updated on 22/Sep/22

$$x^3+\frac{1}{x^3}=1$$$$\frac{{(x^5+\frac{1}{x^5})}^3-1}{x^5+\frac{1}{x^5}}=?$$$$\text{You can't use 'macro parameter character \#' in math mode}$$

Answered by Rasheed.Sindhi last updated on 22/Sep/22

$$\overline{)\begin{array}{c}{x}^3+\frac{1}{x^3}=1\end{array}}$$$${(x^3+\frac{1}{x^3})}^3={\left(1\right)}^3$$$$x^9+\frac{1}{x^9}+3(x^3+\frac{1}{x^3})=1$$$$x^9+\frac{1}{x^9}+3\left(1\right)=1$$$$x^9+\frac{1}{x^9}=1-3=-2\overline{)\begin{array}{c}{x}^9+\frac{1}{x^9}=-2\end{array}}$$$${(x^3+\frac{1}{x^3})}^5=x^{15}+\frac{1}{x^{15}}+5(x^9+\frac{1}{x^9})+10(x^3+\frac{1}{x^3})$$$${\left(1\right)}^5=x^{15}+\frac{1}{x^{15}}+5(-2)+10\left(1\right)$$$$x^{15}+\frac{1}{x^{15}}=1\overline{)\begin{array}{c}{x}^{15}+\frac{1}{x^{15}}=1\end{array}}$$$${(x^5+\frac{1}{x^5})}^3=x^{15}+\frac{1}{x^{15}}+3(x^5+\frac{1}{x^5})$$$$Let{x}^5+\frac{1}{x^5}=u$$$$u^3=3u+1$$$$$$$$\frac{{(x^5+\frac{1}{x^5})}^3-1}{x^5+\frac{1}{x^5}}=\frac{u^3-1}{u}=\frac{3u+1-1}{u}=3$$

Commented by Tawa11 last updated on 23/Sep/22

$$\mathrm{Great}\mathrm{sir}$$

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