Assuming that your masses are really supposed to be in scientific notation, that is to say that the (1.9*1027) kg should really be (1.9*10^27) kg and the (8.9*1022) kg should really be (8.9*10^22) kg, which is more consistent with reality, the methods to solve these are as follows.
Part A: The force of gravity acting on Io can be found using Newtons Law of Gravitation, the Gravitational Force Equation:
F = G * (m1 * m2) / (r^2)
Where 'G' is Newton's gravitational constant (6.67 * 10^(-11)) with units N*m^2/kg^2, 'm1' is the mass of object one in kilograms, 'm2' is the mass of object 2 in kg, and 'r' is the distance between them in meters.
It is first important to realize that the distance between the masses is given in kilometers. We must first convert this value into the required units of the Gravitational Force Equation; 421,700 km becomes 421,700,000 m. After plugging these values into a calculator to solve for F,
(6.67*10^-11) * ( (1.9*10^27) * (8.9*10^22) ) / (421700000^2) becomes
Force or gravity between Jupiter and Io = (6.343*10^22) N
Part B: Acceleration due to force is Newtons second law of motion:
F = m*a
Where 'F' is the force exerted on the object in Newtons, 'm' is the mass of the object in kg, and 'a' is the acceleration of that object in m/s^2. To solve for the acceleration, divide both sides by 'm' to produce the equation:
F/m = a
Plugging in the known values, the force of gravity from part A and the mass of Jupiter, we attain
(6.67*10^22) / (1.9*10^27) = a
(3.3*10^-5) = a, the acceleration of Jupiter due to the force of gravity produced by Io.
Acceleration of gravity can be found using the gravitational acceleration equation:
a = G * m / r^2
where G is the same Newtonian Gravity constant (6.67*10^-11) with units N*m^2/s^2, m is the mass of the large object, and r is the distance from the point-mass center of the object.
At this point, it is important to convert the radius of Io from kilometers to meters; 1,821 km becomes 1,821,000 m. After substituting these known values, the equation becomes
a = (6.67*10^-11) * (8.9*10^22) / (1821000^2)
a = 1.79
The acceleration of gravity on Io, 'a', is 1.79 m/s^2
The change in displacement after some time due to acceleration of gravity can be found using kinematics equations. The particular kinematics equation useful for solving the problem is:
Change in y = Vi*t + (1/2)*a*t^2
Where 'Change in y' represents the change in height in meters, 'Vi' represents the initial velocity of the object in m/s, 't' represents the corresponding time in seconds, and 'a' represents the acceleration of the object in m/s^2. We know the initial velocity is 0 because we hold it before dropping it the rock, we know that vertical acceleration due to gravity on Io is 1.79 m/s^2 from part C, and we know that the displacement of the rock is a vertical 3 meters. Substituting known values yields the equation:
3 = 0*t + (1/2)* (1.79)*t^2
Solving for 't' yields
t = +/- (6/1.79)^(1/2)
t = 1.83 seconds for the rock to hit the ground.
I hope these help you!