Chapter 1: Where is the arcade going?

My grandma like a small child on Christmas morning painfully awaited for an earthquake to see where the arcade is going. Then it happened, the ground shook, placing herself under the kitchen table my grandma grinned uncontrollably, the anticipation was even more unbearable. She ran out the door and measured how far the arcade had moved. She sent me the following image with this letter:

"The arcade has moved 3.36 miles from where it was before. I have explored a couple of things with the equation xn + yn = 1 and I have written separetly why n cannot be zero. So unfortunetly I will never know where the arcade was before I began my observations. Since this equation has worked so far for me, I decided to use geometry sketchpad to make a conjecture on the movement of the arcade. I created another image with the n being from 1 to 10 and I was able to make the following chart:

n
Distance Moved in Miles
2
3.36
3
1.40
4
0.77
5
0.48
6
0.33
7
0.24
8
0.18
9
0.14
10
0.12