To find the radius of a geo-stationary satellite in terms of R_e, we see---

M_s*w^2*R_s=G*M_eM_s/R_s^2 from where using GM_e=g R_e^2,

We get R_s^3=gR_e^2/w^2. w can be evaluated. The problem is because of the fact that
Kleppner provides the soln. R_s=6.6R_e.How, one can get rid of fractional exponent?

What? As far as I can recall, there is no fractional exponent.

when you directly calculate the R_s from the relation, an exponent of 2/3 appears on R_e

Ms w² Rs = G Me Ms / Rs² ... (1)

G Me = g Re² ... (2)

Dividing (2) by Rs² gives:

G Me / Rs² = g (Re/Rs)² ... (3)

Plugging (3) into (1) and dividing by Ms gives:

w² Rs = g (Re/Rs)² ... (4)

Re-arranging to find Rs, we get:

Rs³ = g (Re / w)²

I assume the question gives you the values of w and g.