what is latest firmware update for airport extreme?

Open AirPort Utility on your Mac (Finder > Applications > Utilities > AirPort Utility)

Click on the picture of the AirPort and another smaller window will appear with information about the AirPort

Hover your mouse over the name of the AirPort to see which particular AirPort model that you have

To check on the firmware that is installed on the AirPort, look for version at the bottom of the window

The "latest" version is either 7.8.1 or 7.9.1 depending on what model of AirPort that you have

However, the "latest" version is well over a year old, so it is unlikely that there is a new update available for your device. If there is, you will see an Update button. Click on the Update button to install new firmware on the AirPort

If you don't see an Update button, then your AirPort already has the most up to date version of firmware.

Make sure that the modem is working correctly by connecting your computer directly to the modem using an Ethernet cable connection. If the modem is working correctly, you will be able to get on the Internet with your computer.

Post back on your results.