The perfect spot between the Earth and the Moon

[Calpalned]

Homework Statement

There exists a spaceship orbiting between the Moon and the Earth. Find the perfect spot where the gravity due to Earth is equal to the one caused by the Moon. I have already solved for the ratio $\frac {R_E}{\sqrt{M_E}} = \frac {R_M}{\sqrt{M_M}}$. $R_E$ is the distance between the spaceship and the Earth while $R_M$ represents the distance between the ship and the Moon. Likewise $M_M$ and $M_E$ are masses. The next step in my solutions guide is $R_E = \frac {(R_E + R_M)\sqrt{M_E}}{\sqrt{M_E}+\sqrt{M_M}}$. I don't understand how they got to that.

Homework Equations

1) Correct answer $= R_E = 3.46 * 10^8$ meters
2) Mass of the Moon $=7.347*10^22$
3) Mass of the Earth $=5.98*10^24$
4) Distance between Moon and Earth $= R_E + R_M = 3.844*10^8$

The Attempt at a Solution

Using ratio I solved for in "The problem statement, all..." I tried to isolate $R_E$
$R_E = \frac{R_M\sqrt{M_E}}{\sqrt{M_M}}$
From 4) from part 2 "Relevant equa..." I get that $R_M = 3.844*10^8 - R_E$
Therefore $R_E = \frac{(3.844*10^8 - R_E)\sqrt{M_E}}{\sqrt{M_M}}$
Thus $R_E = (3.844*10^8)(\frac{\sqrt{M_E}}{\sqrt{M_M}}) - R_E(\frac{\sqrt{M_E}}{\sqrt{M_M}})$
$R_E + R_E(\frac{\sqrt{M_E}}{\sqrt{M_M}}) = (3.844*10^8)(\frac{\sqrt{M_E}}{\sqrt{M_M}})$
$R_E(1+(\frac{\sqrt{M_E}}{\sqrt{M_M}})) = (3.844*10^8)(\frac{\sqrt{M_E}}{\sqrt{M_M}})$
$R_E = \frac{(3.844*10^8)(\frac{\sqrt{M_E}}{\sqrt{M_M}})}{1+(\frac{\sqrt{M_E}}{\sqrt{M_M}})}$
While I did get the correct answer, my equation is so complicated. How do I turn it into the one given by the solutions manual?

 

[Calpalned]

Latex refuses to load for some reason... Any solutions? Thanks everyone.

 

[PeroK]

Apart from using that specific number instead of $R_E+R_M$, there is nothing different about your solution. All you have to do is multiply top and bottom by $\sqrt{M_M}$ to get the book answer.

 

[Calpalned]

Apart from using that specific number instead of $R_E+R_M$, there is nothing different about your solution. All you have to do is multiply top and bottom by $\sqrt{M_M}$ to get the book answer.

Now I see it! Thank you so much!