This is the exact problem from the paper and I got an answer of 4.2 * 10^7 and I was wondering if this was the right answer.

What is the height above the earth's surface at which all synchronous satellites (regardless of mass) must be placed in orbit? The period of synchronous satellites is one day. The radius of the earth 6.36(10)6m

That is high. Did you work if from the Earth's surface, for from the center of Earth?

where is m v^2/r = G M m/r^2

as you said, m cancels
v^2 = G M/r
T = 24 h *3600 s/h = 86,400 s
2 pi r = v T = 86,400 v
so
r = 13,751 v
v^3 = G M/13,751 = 6.67*10^-11 * 6*10^24/1.38*10^4
v^3 = 29*10^9
v = 3.07 * 10^3
then r = 4.24 * 10^7
eight = (10^7 )(4.24 - .636) = 3.6 *10&
remember to subtract the radius of earth to get the altitude.

I worked it from the center of the Earth so i have to go back and subtract the radius of the Earth from that answer.

To determine the height above the Earth's surface at which synchronous satellites must be placed, we can use the formula for the period of a satellite's orbit, which is related to the orbital radius.

The period T of a satellite is given by the equation:

T = 2π * √(r^3 / GM)

Where:
T = period of orbit in seconds
r = orbital radius from the center of the Earth in meters
G = gravitational constant (approximately 6.674 × 10^-11 m^3 kg^-1 s^-2)
M = mass of the Earth (approximately 5.97 × 10^24 kg)

In this case, the period T of synchronous satellites is one day, which is equivalent to 86,400 seconds. We can substitute these known values into the equation:

86,400 = 2π * √(r^3 / (6.674 × 10^-11 * 5.97 × 10^24))

To solve for the orbital radius r, we can rearrange the equation:

r^3 = (86,400^2) * (6.674 × 10^-11 * 5.97 × 10^24) / (2π)^2

r^3 = (86,400^2) * (6.674 × 10^-11 * 5.97 × 10^24) / (4π^2)

Taking the cube root of both sides, we can calculate the orbital radius:

r = ∛[(86,400^2) * (6.674 × 10^-11 * 5.97 × 10^24) / (4π^2)]

By substituting the values given into the equation and evaluating it, we can find the correct answer for the height above the Earth's surface at which synchronous satellites must be placed.