[Factorization] [Exponential Equations]

Find all positive roots of the equation $x^{x}+x^{1-x}=x+1$.

#
Since $x>0$, we have $0=x^{2x}+x-x^{x+1}-x^{x}=x^{x}(x^{x}-1)-x(x^{x}-1)=x(x^{x}-1)(x^{x-1}-1)$. Therefore, $x=1$.

## Answer

$x=1$.

## [ Numerical tables and their properties ]

Problem