Containing an "exponent" does not make a trend exponential. It makes it a power factor, or N order polynomial.
An exponential function has the variable AS an exponent. An n order power function has the variable to the power of n. Y = A*e^X is indeed an exponential function, because the variable, X, is in the exponent. Y = X^2, a second order polynomial POWER FUNCTION, is NOT an exponential function, as the variable is NOT a component of the exponent.
Yes, indeed you can. Functions including only fixed exponents - aka constant exponents - are indeed NOT exponential functions. That's the very definition of exponential functions as compared to that of power functions.
Go back to high school.