The mass of the earth is $6 \times {10^{24}}$ kg and that of the moon is $7.4 \times {10^{22}}$ kg. If the distance between the earth and the moon is $3.84 \times {10^5}$ km, calculate the force exerted by the earth on the moon. Consider $G = 6.7 \times {10^{ - 11}}{\text{N}}{{\text{m}}^{\text{2}}}{\text{k}}{{\text{g}}^{{\text{ - 2}}}}$.

Hint
Newton's universal law of gravitation can be used to find the gravitational force between any two objects. This force depends on the mass of the objects involved and the distance between them.
$\Rightarrow F = G\dfrac{{Mm}}{{{r^2}}}$ where $F$ is the force of attraction between two objects of mass $M$and $m$, separated by a distance $r$. $G$ is the universal gravitational constant.

Complete step by step answer
The two bodies specified in the question are the Earth and the moon. We are asked to find the force exerted by the Earth on the moon. The data provided to us include:
Mass of the Earth $M = 6 \times {10^{24}}$kg
Mass of the moon $m = 7.4 \times {10^{22}}$kg
Distance between the two $r = 3.84 \times {10^5}$ km $ = 3.84 \times {10^8}$ m [As 1 km = 1000 m]
Gravitational constant $G = 6.7 \times {10^{ - 11}}{\text{N}}{{\text{m}}^{\text{2}}}{\text{k}}{{\text{g}}^{{\text{ - 2}}}}$
We are aware that the force between two gravitationally bound objects can be found by applying Newton’s Law as:
$\Rightarrow F = G\dfrac{{Mm}}{{{r^2}}}$
Substituting the given values in this equation gives us:
$\Rightarrow F = 6.7 \times {10^{ - 11}}\dfrac{{6 \times {{10}^{24}} \times 7.4 \times {{10}^{22}}}}{{{{(3.84 \times {{10}^8})}^2}}}$
Expanding the square term and cancelling all the powers of 10 gives us:
$\Rightarrow F = \dfrac{{6.7 \times 6 \times 7.4 \times {{10}^{35}}}}{{14.7456 \times {{10}^{16}}}} = \dfrac{{297.48}}{{14.746}} \times {10^{19}}$
This gives us the force between the two as:
$\Rightarrow F = 20.17 \times {10^{19}}$ N

Note
In the question we were asked to find the force exerted by the Earth on the moon, but this is also equal to the force exerted by the moon on Earth. Newton’s third law of motion states that every force has an equal and opposite reaction. That is why there is no distinction between equations when trying to find the force, and just one equation is valid no matter which body exerts force on the other.