the quantity of a radioactive substance decays according to the function $q(t)=100e^{-\frac{t}{4}}$, where…

Explanation:

Step1: Recall exponent rules

Use the rule $(a^m)^n=a^{mn}$.

Step2: Analyze option C

For $Q(t) = 100(e^{-2})^{\frac{t}{8}}$, by the exponent - rule $(a^m)^n=a^{mn}$, we have $(e^{-2})^{\frac{t}{8}}=e^{-2\times\frac{t}{8}}=e^{-\frac{t}{4}}$. So $Q(t)=100(e^{-2})^{\frac{t}{8}} = 100e^{-\frac{t}{4}}$.

Step3: Analyze option D

For $Q(t)=100(e^{-\frac{1}{4}})^t$, by the exponent - rule $(a^m)^n=a^{mn}$, we have $(e^{-\frac{1}{4}})^t=e^{-\frac{1}{4}t}=e^{-\frac{t}{4}}$. So $Q(t)=100(e^{-\frac{1}{4}})^t = 100e^{-\frac{t}{4}}$.

Step4: Analyze other options

• Option A: $Q(t)=-25e^t$ is not equivalent to $100e^{-\frac{t}{4}}$ since the coefficient and the exponent of $e$ are different.
• Option B: $Q(t)=25e^{-t}$ is not equivalent as the coefficient and the exponent of $e$ are different.
• Option E: $Q(t)=100(-\frac{e}{4})^t$ is not equivalent because of the negative sign and the fraction in the base.
• Option F: $Q(t)=400e^{-t}$ is not equivalent due to the coefficient and the exponent of $e$.

Answer:

C. $Q(t)=100(e^{-2})^{\frac{t}{8}}$, D. $Q(t)=100(e^{-\frac{1}{4}})^t$