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Question Number 158187 by HongKing last updated on 31/Oct/21

$$\mathrm{Solve}\mathrm{for}\mathrm{real}\mathrm{numbers}:$$$$\{\begin{array}{l}\sqrt{\mathrm{x}}-{\mathrm{y}}^5=3\\ \sqrt[5]{\sqrt{\mathrm{x}}-3}-\sqrt[5]{{\mathrm{y}}^5+6}=-1\end{array}$$$$$$

Answered by MJS_new last updated on 31/Oct/21

$$\sqrt{x}=y^5+3[x⩾0\wedge \sqrt{x}⩾0]$$$$y-{(y^5+6)}^{1/5}=-1$$$$y^4+2y^3+2y^2+y-1=0$$$$(y^2+y+\frac{1-\sqrt{5}}{2})(y^2+y+\frac{1+\sqrt{5}}{2})=0$$$$y=\frac{-1+\sqrt{-1+2\sqrt{5}}}{2}$$$$\Rightarrow x=\frac{7+5\sqrt{5}}{2}$$

Commented by HongKing last updated on 31/Oct/21

$$\mathrm{perfect}\mathrm{dear}\mathbf{S}\mathrm{er},\mathrm{thank}\mathrm{you}\mathrm{so}\mathrm{much}$$

Commented by ajfour last updated on 01/Nov/21

$$awestrikeprecision!$$

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