how does the space shuttle maneuver in the exosphere

It uses small use electrical-ion or monopropellant chemical thrusters

The space shuttle, or any spacecraft, does not maneuver in the exosphere because the exosphere is the outermost layer of Earth's atmosphere, which extends from about 500 kilometers (311 miles) above the surface of the Earth. Space shuttles typically maneuver and operate in the thermosphere and sometimes the mesosphere.

To maneuver a space shuttle once it reaches space, it primarily relies on the use of onboard rocket engines and thrusters. These engines provide the necessary thrust to change the shuttle's speed, direction, and altitude.

Here's a general sequence of maneuvers that a space shuttle might perform in space:

1. Launch: The space shuttle is launched into space using a combination of solid rocket boosters and main engines. During launch, it follows a predetermined trajectory to reach the desired orbital altitude.

2. Circularization burn: Once the space shuttle reaches its intended orbit, it performs a circularization burn. This burn increases its orbital speed to match the required velocity for a stable circular orbit. The engines are fired briefly to achieve this.

3. Orbital maneuvering: To change its orbit or adjust its position within an orbit, the space shuttle uses the Orbital Maneuvering System (OMS) engines. These engines are used to increase or decrease the shuttle's speed, allowing it to either raise or lower its orbital altitude.

4. Docking/Undocking: If the shuttle is required to dock with another spacecraft, such as the International Space Station (ISS), it approaches the target carefully and performs a carefully timed docking maneuver. To undock, the reverse process is followed.

5. Re-entry and Landing: When the mission is complete, the space shuttle prepares for re-entry into the Earth's atmosphere. It relies on its reaction control system (RCS) thrusters to control its attitude and make adjustments during re-entry. Once it re-enters the atmosphere, it glides back to Earth for a landing on a runway like an airplane.

So, in summary, the space shuttle maneuvers in space primarily using rocket engines during various stages of its mission, including launch, orbital adjustment, and re-entry.

The Space Shuttle Orbiter travels ~13,875 miles before reaching a typical 185 mile altitude circular orbit, approximately 25 miles above the oft-defined 160 miles altitude of the lower boundry of the exosphere. The Space Shuttle Reaction Control System provides attitude control and three axis translation during orbit insertion, on-orbit, and reentry phases of flight.

the only rocket propellants remaining on the Space Shuttle Orbiter once it has reached orbit are the propellants associated with the Orbital Maneuvering System (OMS) and the Reaction Control System (RCS).
The RCS provides for attitude control and three axis translation during the orbit insertion, on-orbit, and reentry phases of each flight as well as primary flight control above 70,000 ft. altitude. The OMS provides the thrust for orbital insertion, orbit circularization, orbit transfer, rendezvous and deorbit.
The OMS system consists of two pods, one on each side of the upper aft fuselage flanking the vertical stabilizer of the Orbiter. Each pod contains a high-pressure helium storage bottle, four tank pressure regulators and controls, a fuel tank, an oxidizer tank, and a pressure fed regeneratively cooled rocket engine. A maximum
of 4,505 lbs. of Monomethyl Hydrazine MMH fuel and 7,433 lbs. of Nitrogen Tetroxide N204 oxidizer can be carried in each pod. Each engine produces 26,700 lbs. of thrust in a vacuum and is designed to be reuseable for 100 missions, and capable of 1,000 starts and 15 hours of cummulative firing.
The Space Shuttle Orbiter does its turning by means of a Reaction Control System (RCS) which has 38 bipropellant primary thrusters and six vernier thrusters to provide attitude control and three axis translation during the orbit insertion, on-orbit, and reentry phases of the flight. The RCS is used as the primary flight control system above 70,000 feet. The RCS consists of three modules, one in the nose of the forward fuselage section, and one in each of the Orbital Maneuvering Syatem (OMS) pods on either side of the vertical tail at the rear of the fuselage. All the modules are used during external tank separation, orbital insertion and orbital maneuvers while only the aft modules are used for reentry. Each of the primary thrusters provides 870 pounds of vacuum thrust and is designed to be reusable for 100 missions, capable of sustaining 50,000 starts and 20,000 seconds of
cumulative firing. The vernier thrusters each provide 25 pounds of vacuum thrust, are designed for 100 missions, being started 500,000 times, for a total of 125,000 seconds of cumulative firing. The RCS enables the orbiter to perform pure rotation about all three axes and pure translation in either direction along all three axes. The propellants used are Monomethyl Hydrazine (MMH) as the fuel and Nitrogen Tetroxide (N204) as the oxidizer. Each of the 3 fuel tanks carries 930 lbs. of MMH and each of the 3 oxidizer tanks carries 1,488 lbs. of N204. The propellants are hypergolic, which means they ignite on contact with one another and need no ignitor.

Ref: Aerofax Datagraph #5-Rockwell International Space Shuttle by Dennis R. Jenkins, Aerofax, Inc., Arlington,
TX, 1989.