Computing the Value of n for algorithmic equation

Question

I am working on This question. Have Solved many of them But stuck with the Computation of n^n = 10^7.. How should i Evaluate above for the value of n.. The original Question is :-

Assume you have two computers, CA and CB, capable of performing 10^7 and 10^9 operations per second, respectively. Both computers run a set of algorithms whose precise complexities f(n) are given below. Determine the size n of the biggest input that can be processed in 1 second for each computer, as in the example.

The precise complexity tells you how many operations are performed to solve an instance of size n. Assume each operation takes the same time and that the input sizes are natural numbers 1, 2, 3, . . . The point of this exercise is to see how much can we gain by going from CA to CB.

Any comments will be appreciated for the evaluating n^n = 10^7 .... Thanks

Answer 1

We have number of operations (per second).

The function maps input size to number of operations.

So all we need to do is set that function equal to the number of operations, and calculate n.

sqrt(n) = 10⁷

sqrt(n)² = (10⁷)²

n = 10¹⁴