This manipulate demo shows how the power and constant behave together when we fix the right endpoint (as our data fitting must -- the graphs of both functions will almost pass through the last point).

Manipulate[
Module[
{},
p2=Plot[Abs[t-c]^p/tend^p,{t,0,tend}]
]
,{{tend,15},5,100,5}
,{{c,0},0,tend,5}
,{{p,2},.05,3,.01}
]

The following two final calculations are supposed to be about equal:

Clear[p]
Series[(((1970+t))^p),{t,0,3}]
p1970=208.81726363472885` 
a1970= 2.1046578223604923357380631343`12.757523227008011*^-684
(1970^(-1+p1970)) p1970*a1970

Clear[p]
Series[(((70+t))^p),{t,0,3}]
p70=8.353923979296233` 
a70=6.832240648605592`*^-12
(70^(-1+p70)) p70*a70

The following pairs of ratios should be about equal:

p1970/p70
1970./70.

a1970/a70
(70^p70)/(1970^p1970)