Let's analyze the expressions for Eric and Andrea in the problem and break down each component.

Eric's Expression: $10 \left( \frac{1}{4} \right)^w$

1. 10: This represents the initial amount of the substance that Eric starts with, which is 10 milligrams.
2. $\frac{1}{4}$: This represents the rate of decay per week, indicating that each week only one-fourth of the previous week's amount remains.
3. $w$: This is the number of weeks that have passed.
4. $\left( \frac{1}{4} \right)^w$: This represents the remaining fraction of the substance after $w$ weeks, as the amount decreases by a factor of $\frac{1}{4}$ each week.

Andrea's Expression: $(1 - 0.2)^w$

1. 1: Represents the full amount before decay is considered.
2. 0.2: This represents the decay rate per week (20%).
3. $1 - 0.2$: Represents the remaining portion of the substance each week, meaning 80% (or 0.8) remains each week.
4. $w$: This is the number of weeks.

These drop-down explanations should help to clarify each part in terms of its role in modeling the exponential decay of the radioactive substances in Eric's and Andrea's cases.

Would you like a more detailed explanation or have questions on how exponential decay works?

Here are some related questions to explore:

1. What would the expressions look like if Eric’s decay rate was changed to one-third per week?
2. How would Andrea’s expression change if her substance decayed by 30% per week instead of 20%?
3. What is the difference in the decay rates of Eric’s and Andrea’s substances after 5 weeks?
4. Can you rewrite Andrea’s expression in terms of a base of 0.8 instead of $1 - 0.2$?
5. How can you determine the remaining substance if you know the decay rate but not the initial amount?

Tip: In exponential decay, the base of the exponential (like $\frac{1}{4}$ or $0.8$) shows what fraction of the substance remains after each unit of time.