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Question Number 201200 by Calculusboy last updated on 01/Dec/23

Commented by Rasheed.Sindhi last updated on 02/Dec/23

${({(x - 1)}^{x - 1})}^{1 / 3} = {(x - 1)}^{{(x - 1)}^{1 / 3}}$
Commented by Rasheed.Sindhi last updated on 02/Dec/23

$S y m m e t r y i n a b o v e q u e s t i o n!$
	Answered by cortano12 last updated on 02/Dec/23

	${(x - 1)}^{\frac{x - 1}{3}} = {(x - 1)}^{\sqrt[3]{x - 1}}; x \neq 1$ $(1) x - 1 = - 1 \Rightarrow x = 0$ $(2) x - 1 = 1 \Rightarrow x = 2$ $(3) \frac{x - 1}{3} = \sqrt[3]{x - 1} \Rightarrow (x - 1) = 3 {(x - 1)}^{1 / 3}$ ${(x - 1)}^{1 / 3} ({(x - 1)}^{2 / 3} - 3) = 0$ ${(x - 1)}^{2 / 3} = 3 \Rightarrow {(x - 1)}^{2} = 27$ $x = 1 \pm 3 \sqrt{3}$
	Commented by Calculusboy last updated on 02/Dec/23

	$t h a n k s s i r$
	Answered by esmaeil last updated on 02/Dec/23
	$((x−1)/3)ln(x−1)=((x−1))^(1/3) ln(x−1)→^(x−1=u) u=3(u)^(1/3) → { ((x=1)),((x=1±3(√3))) :}$
	$\frac{x - 1}{3} l n (x - 1) = \sqrt[3]{x - 1} l n (x - 1) \overset{x - 1 = u}{\to}$ $u = 3 \sqrt[3]{u} \to {\begin{cases} x = 1 \\ x = 1 \pm 3 \sqrt{3} \end{cases}$
	Commented by Calculusboy last updated on 02/Dec/23

	$t h a n k s$
	Answered by Rasheed.Sindhi last updated on 02/Dec/23
	$(x−1)^((x−1)/3) =(x−1)^((x−1)^(1/3) ) (x−1)^(((x−1)/3)−(x−1)^(1/3) ) =1=(x−1)^0 { ((x−1=1⇒x=2✓)),((x−1=−1⇒x=0✓)),((((x−1)/3)−(x−1)^(1/3) =0)) :} ((x−1−3(x−1)^(1/3) )/3)=0 x−1=3(x−1)^(1/3) (x−1)^3 =27(x−1) (x−1)^3 −27(x−1)=0 (x−1)( (x−1)^2 −27)=0 x≠1⇒ x−1=±3(√3) ⇒x=1±3(√3) ✓$
	${(x - 1)}^{\frac{x - 1}{3}} = {(x - 1)}^{{(x - 1)}^{1 / 3}}$ ${(x - 1)}^{\frac{x - 1}{3} - {(x - 1)}^{1 / 3}} = 1 = {(x - 1)}^{0}$ ${\begin{cases} x - 1 = 1 \Rightarrow x = 2 ✓ \\ x - 1 = - 1 \Rightarrow x = 0 ✓ \\ \frac{x - 1}{3} - {(x - 1)}^{1 / 3} = 0 \end{cases}$ $\frac{x - 1 - 3 {(x - 1)}^{1 / 3}}{3} = 0$ $x - 1 = 3 {(x - 1)}^{1 / 3}$ ${(x - 1)}^{3} = 27 (x - 1)$ ${(x - 1)}^{3} - 27 (x - 1) = 0$ $(x - 1) ({(x - 1)}^{2} - 27) = 0$ $x \neq 1 \Rightarrow x - 1 = \pm 3 \sqrt{3} \Rightarrow x = 1 \pm 3 \sqrt{3} ✓$
	Commented by Calculusboy last updated on 02/Dec/23

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