You're getting a bit confused again.
Your laptop output
is 20v DC @ 9amps, which is 180watts (and a mighty HUGE power supply, might I add. That's just simply insane!). Because it's 9amps at 20 volts DC, it would never come close to pulling 9 amps at 120v. What is going on is basically a transformer. It's taking a high voltage at a low current draw and in transforming it into a low voltage at a high current, which is fine. Over a short distance, low voltage high current works fine. It's over spans that it creates a problem. Anyhow, i'm not going to get into that right now.
With laptops you really need not worry about power surges. They are so regulated that it's really not an issue. Keep in mind, 20v @ 9amps is at MOST what the laptop will ever draw. That's charging the battery from a dead state, the cpu running at full bore, both drives actively going, backlight on high, etc etc. For the most part, it's going to be operating on less power, especially if the battery is already charged. Case in point, girlfriends HP. If the battery is removed from the unit or if the battery is fully charged, even with everything else going full bore the lappy is only drawing ~65 watts (out of a 90 watt supply). Under normal use it's only drawing 1\2 of what the adapter can put out. So, if it's drawing ~50 watts DC out of a 90watt DC supply, it's really only pulling ~110 AC
out of the possible 216 that my paticular adapter is rated for.
Wow, i'm really babbling here...
But what it all boils down to is, you need an inverter that can put out atleast 350 watts, but again I would side on a bit more power. I would go for ATLEAST 400, but 450 or 500 would certainly fair better. Of course, then you get into what the inverter is drawing off of the car electrical system, because don't forget, you're taking DC power from the car, converting it to AC with the inverter, then the laptop power supply is converting that AC back to DC again! It's the utmost in inefficency
Figure a 500w AC inverter is going to draw somewhere around 50 amps DC off of the car electrical system. It certainly needs to be direct wired to the battery as most car cigarette lighters are only rated for a 20 amp circuit. I would recommend using a 6 or 4 gauge cable to go from the battery to the inverter.