Me and my friend are new to linear algebra (we posted earlier, thanks for the help) and we're a little confused on another question here:

How do we solve this with that big scary exponent hanging around in the corner there? Doing it manually would take ages, is there some specific method to solving with large exponents? My friend said she googled it and said something about a "eigen value" or something - we haven't learned that yet.

Can anyone provide insight into solving this? Sorry if we're bugging you all, we've managed to get most of this stuff down except this one puzzled us.

Thanks a lot!