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Question Number 202984 by hardmath last updated on 06/Jan/24

$$\mathrm{If}{\mathrm{x}}^{{\mathbf{x}}^3}=3$$$$\mathrm{Find}\mathrm{x}=?$$

Commented by mr W last updated on 07/Jan/24

$$generally$$$$if{x}^{x^{..{.}^{x^a}}}=a>0$$$$\Rightarrow x=\sqrt[a]{a}$$$$withanynumberoflevelsofthe$$$$powertower.$$

Commented by hardmath last updated on 08/Jan/24

$$\mathrm{thank}\mathrm{you}\mathrm{dear}\mathrm{professor}$$

Answered by Frix last updated on 06/Jan/24

$$\mathrm{Simply}\mathrm{let}x={\mathrm{e}}^{\frac{t}{3}}\Rightarrow$$$${\mathrm{e}}^{\frac{{\mathrm{e}}^{t}t}{3}}=3$$$${\mathrm{e}}^{t}t=3\ln 3$$$$t=\ln 3$$$$x=\sqrt[3]{3}$$

Commented by Frix last updated on 06/Jan/24

$$x^{x^n}=n\Rightarrow x=\sqrt[n]{n}$$

Answered by EricAsaah last updated on 06/Jan/24

$${\left(x^{x^3}\right)}^3=3^3$$$$x^{3x^3}=3^3$$$$(x^{{3)}^{x^3}}=3^{3}Let{x}^3=t$$$$t^t=3^3$$$$\to t=3$$$$\therefore x^3=3$$$$x{=}^3\sqrt{3}$$$$$$

Commented by hardmath last updated on 08/Jan/24

$$\mathrm{thank}\mathrm{hou}\mathrm{dear}\mathrm{professors}$$

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