Answer
At a distance of $~~3.44\times 10^8~m~~$ from Earth, the net gravitational force on the spaceship is zero.
Work Step by Step
The gravitational force on the spaceship due to the moon is directed toward the moon.
The gravitational force on the spaceship due to the Earth is directed toward the Earth.
For the net gravitational force on the spaceship to be zero, the gravitational force on the spaceship due to the moon must be equal in magnitude to the gravitational force on the spaceship due to the Earth.
Let $r$ be the distance from the Earth to the spaceship.
Then $~~(3.82\times 10^8~m-r)~~$ is the distance from the moon to the spaceship.
We can find $r$:
$\frac{G~m_E~m_s}{r^2} = \frac{G~m_M~m_s}{(3.82\times 10^8-r)^2}$
$m_E~(3.82\times 10^8-r)^2 = m_M~r^2$
$\sqrt{m_E}~(3.82\times 10^8-r) = \sqrt{m_M}~r$
$(\sqrt{m_M}+\sqrt{m_E})~r = \sqrt{m_E}~(3.82\times 10^8)$
$r = \frac{\sqrt{m_E}~(3.82\times 10^8)}{\sqrt{m_M}+\sqrt{m_E}}$
$r = \frac{\sqrt{5.98\times 10^{24}}~(3.82\times 10^8)}{\sqrt{7.36\times 10^{22}}+\sqrt{5.98\times 10^{24}}}$
$r = (0.900)(3.82\times 10^8~m)$
$r = 3.44\times 10^8~m$
At a distance of $~~3.44\times 10^8~m~~$ from Earth, the net gravitational force on the spaceship is zero.
