Can I run this software on a 2010 macbook air?

The system requirements on Apple Website for FCP X states the following:

• Mac computer with an Intel Core 2 Duo processor or better.
• 2GB of RAM (4GB of RAM recommended).
• OpenCL-capable graphics card or
  Intel HD Graphics 3000 or later.
• 256MB of VRAM (512MB of VRAM recommended).
• Display with 1280-by-768 resolution or higher.
• OS X v10.6.8 or OS X v10.7.4 or later (OS X v10.7.4 or later for broadcast monitoring).
• 2.4GB of disk space.

If this is what you have as a Minimum then it should be fine. I still would run FCP X on a more powerful machine than a MacBook Air though.

Hope this is helpful.

The simplest way to find out is to download a trial version

FCPX works on my mba 11 core 2 duo 1.6Ghz