The number of positive real roots of the equation $x^6-2x-3=0$  is

The given equation is $f\left(x\right)=x^6-2x-3=0$

It has only one change of sign,

so there may be at the most one positive real root.

Also $f\left(x\right)$  is of even degree with constant term negative.

 So there must be at least two real roots, one positive and one negative.

 Thus we see that there is at least one positive real root and not more than one

Hence $f\left(x\right)=0$ has only one positive real root.