Note that you don't need to work out the numbers exactly to decide which is bigger!

Some possible methods:

1) Can you split up the powers into smaller pieces, or consider another quantity which your calculator can handle from which you can deduce the answer?

or 2) Can you use inequalities and logic such as 'if X > Y and Y > Z then we know that X > Z'

or 3) What happens when you raise $100^{1/100}$ to the power $9900$?

Some possible methods:

1) Can you split up the powers into smaller pieces, or consider another quantity which your calculator can handle from which you can deduce the answer?

or 2) Can you use inequalities and logic such as 'if X > Y and Y > Z then we know that X > Z'

or 3) What happens when you raise $100^{1/100}$ to the power $9900$?