show that the circles x^2 + y^2 - 10x - 8y + 18 = 0 and x^2 + y^2 - 8x - 4y + 14x = 0 do not intersect.
Sorry. There's still a typo. Now you have two x terms. Don't you read what you post?
As typed, your two circles do intersect.
http://www.wolframalpha.com/input/?i=plot+x%5E2+%2B+y%5E2+-+10x+-+8y+%2B+18+%3D+0+,+x%5E2+%2B+y%5E2+-+8x+-+4y+%2B+14x+%3D+0
Rather than just reposting again, why not do what I showed you earlier? Complete the squares to find the two centers.
Then find the distance between the centers, and show that it is too far for the two radii to reach.