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Archimedes and Numerical Roots

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

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What Do Functions Do for Tiny X?

Looking at small values of functions. Motivating the existence of the Taylor expansion.

Approximately Certain

Age 14 to 18 Challenge Level:
How could you estimate each of the quantities? What extra information might you need to find out?
For some questions, you may need to be more accurate in your estimations than others to arrange the list in order of magnitude.