the amount of radioactive element remaining, ...

# Explanation: ## Step1: Recall the general form of exponential - decay The general form of exponential decay is $r = a(1 - p)^d$, where $a$ is the initial amount, $p$ is the daily percent decrease (as a decimal), and $d$ is the number of days. We are given $r = 100(\frac{1}{2})^{\frac{d}{5}}$. Let's rewrite it in the general form. First, rewrite $(\frac{1}{2})^{\frac{d}{5}}$ as $((\frac{1}{2})^{\frac{1}{5}})^d$. Calculate $(\frac{1}{2})^{\frac{1}{5}}=\sqrt[5]{\frac{1}{2}}$. ## Step2: Calculate $(\frac{1}{2})^{\frac{1}{5}}$ $(\frac{1}{2})^{\frac{1}{5}}=\frac{1}{\sqrt[5]{2}}$. Using a calculator, $\sqrt[5]{2}\approx1.1487$, so $(\frac{1}{2})^{\frac{1}{5}}\approx0.8706$. ## Step3: Find the percent - decrease Since $r=a(1 - p)^d$ and $r = 100(0.8706)^d$, then $1-p = 0.8706$. Solving for $p$, we get $p=1 - 0.8706=0.1294$. To convert $p$ to a percentage, multiply by 100. So the daily percent decrease is $12.94\%$. # Answer: 12.94%