Find the radius of a geo-stationary satellite

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The radius of a geo-stationary satellite (R_s) can be determined using the equation R_s^3 = gR_e^2/w^2, where g represents gravitational acceleration, R_e is the Earth's radius, and w is the angular velocity. According to Kleppner's solution, R_s is approximately 6.6 times R_e. The discussion highlights the challenge of eliminating the fractional exponent in the equation, suggesting that numerical substitution may be necessary to derive the 6.6 relationship definitively.

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Kolahal Bhattacharya
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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?
 
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My guess is that you have to plug in the numbers, and then in the end get the 6.6 relationship. I don't see an algebraic way to do it offhand.
 

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