Question

Solve $x^{\sqrt{x}}=(\sqrt{x}{)}^x$

Answer 1

The equation can be re-written as $x^{\sqrt{x}}=x^{x/2}$
By trial , $x=1$ is a solution.
Now equating the exponents , we have
$\sqrt{x}=\frac{1}{2}xorx=\frac{1}{4}{x}^{2}orx(x-4)=0$
Since $x=0$ does not satisfy the given equation ,we get $x=4$
Hence the roots of the given equation are $x=1,4$