The distance from earth to the moon is about 22^4 miles. the distance from earth to neptune is about 22^7 miles. which distance is the greater distance and about how many times is it?

You are NOT using scientific notation correctly.  When you say 22^4 you are saying 22*22*22*22 which equals 234,256.  What you should say is Earth to Moon distance = 22 * 10^4 miles.  (To be really precise, it should be expressed as 2.2 * 10^5 miles).

Earth to Neptune distance = 2.2 * 10^8 miles.

2.2 * 10^8 / 2.2 * 10^5 = 1,000.  So the Earth to Neptune distance is 1,000 times greater than the Earth to Moon distance.