Question

How do you identify equations as exponential growth, exponential decay, linear growth or linear decay `M(t)=8(2)^(1/6t)`?

Answer 1

A linear (decay or growth) has the variable 'in line" with the other terms as in:
`M(t)=at+c`
The constant `a` determines whether it is a decay (`a<0`) or growth (`a>0`).
For example `M(t)=4t+7` where `a=4>0` is a linear growth.

An exponential has the variable up on the "first floor" as in:
`M(t)=ka^(bt)` where `k,a,b` are all constants (with `k>0`).
The constant `b` determines whether it is a decay (`b<0`) or growth (`b>0`).
In your case you have `M(t)=8*(2^(1/6t))` so that it is an exponential (`t` on the first floor) and `b=1/6>0` is an exponential growth.

Graphically:
graph{8(2^(x/6)) [-25.66, 25.66, -12.83, 12.82]}