Functions Modeling Change: A Preparation for Calculus, 5th Edition

Yes, $Q(t)=2^t$ is exponential.
Using the fact that $(a^b)^n=a^{bn}$, this function becomes $$Q(t)=(8^{1/3})^t$$ Using the fact that $8^{1/3}=\sqrt[3]{8}=2$, the function becomes exponential. $$Q(t)=2^t$$