Eight Times Every Doubling

You time one function on random inputs of increasing size:

Assume the running time follows a power law $T(n) \approx a n^b$ . What is $b$ , and how long should $n = 8000$ take?

$b \approx 3$ , but three timings cannot fix $a$ , so no prediction is possible.

$b = 3$ , about 260 seconds.

$b = 2$ , about 130 seconds.

$b = 8$ , about 8,300 seconds.