I'm somewhat confused by the Ranger class attack powers from the Player's Handbook. Many powers require the player to wield two melee weapons. What's bothering me is the damage: the book says the damage for Careful Attack is 1[W]. But what if you are using two different melee weapons? For example, if I'm holding a longsword in my main hand, whose damage is 1d8; and a short sword in my off hand, whose damage is 1d6, which do I use for damage? The attack suggests that you use both weapons for the attack, but the damage is only 1[W], so what do I do?
Direct Math for your question and an explanation about Careful Attack.
Attack: Roll 1d20+to-hit modifier. If you hit roll (1d8 OR 1d6) + damage modifier .
Careful Attack's requirement to have two melee weapons or a ranged weapon is simply that, a requirement. When you make the attack while wielding two melee weapons you can choose which melee weapon you use (since it doesn't specify main-hand or off-hand). Unless your off hand shortsword had a better magical property you would always want to go with the bigger damage die of the mainhand weapon.
Attacks that let you Attack more than once will specify whether or not you simply use the main-hand weapon or you do one for each hand
For example, twin strike specifies that when made as a melee attack it triggers on the main hand and then the off-hand weapons, thus you make 1 attack & damage roll for the main hand and a separate attack and damage roll using your off hand weapon (ranged rangers using Twin Strike simply roll twice for their ranged weapon).
If you're playing a Two-Weapon ranger you should have chosen the Class Feature Two-Blade Fighting Style
This is critical for Two-Weapon rangers because it allows you to have one handed weapons in both hands (without the off-hand property being required). Thus you should instead be using two identical longswords (or better yet if you have a feat to spend, pick up Weapon Proficiency:Bastard Sword. 1d10 high crit one handed weapon).