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The problem involves radioactive decay, which is modeled by the equation:

\[
a = A \cdot \left( \frac{1}{2} \right)^t
\]

where:
- \( a \) is the amount of material at time \( t \),
- \( A \) is the initial amount of the material,
- \( t \) is the time passed (in some units, like years or hours).

You are asked to compare the amount of radioactive material at \( t = 6 \) to the amount at \( t = 2 \).

### Step 1: Amount of material at \( t = 6 \)
Using the formula:
\[
a_6 = A \cdot \left( \frac{1}{2} \right)^6
\]

### Step 2: Amount of material at \( t = 2 \)
Similarly:
\[
a_2 = A \cdot \left( \frac{1}{2} \right)^2
\]

### Step 3: Compare the two amounts
To compare the amounts, we take the ratio of the two expressions:
\[
\frac{a_6}{a_2} = \frac{A \cdot \left( \frac{1}{2} \right)^6}{A \cdot \left( \frac{1}{2} \right)^2}
\]

The \( A \)'s cancel out, leaving:
\[
\frac{a_6}{a_2} = \frac{\left( \frac{1}{2} \right)^6}{\left( \frac{1}{2} \right)^2} = \left( \frac{1}{2} \right)^{6-2} = \left( \frac{1}{2} \right)^4
\]

Since \( \left( \frac{1}{2} \right)^4 = \frac{1}{16} \), the amount of radioactive material at \( t = 6 \) is **1/16th** of the amount at \( t = 2 \).

### Conclusion:
The amount of radioactive material present at \( t = 6 \) is **16 times less** than the amount present at \( t = 2 \).

---

Would you like further clarification or a deeper dive into any part of this solution?

Here are 5 follow-up questions to explore further:
1. What happens to the amount of material if the time \( t \) increases indefinitely?
2. How is the half-life of a material related to this decay equation?
3. What is the significance of the factor \( \frac{1}{2} \) in this decay model?
4. How would the decay process change if the time \( t \) is measured in a different unit (e.g., days instead of years)?
5. How can this model be applied to different materials with varying half-lives?

**Tip**: When comparing rates of change in exponential decay, ratios often reveal insights about how quickly the quantity is diminishing across time intervals.

How does the amount of radioactive material present at time t = 6 compare to the amount present at t = 2?

Learn how to compare the amount of radioactive material present at t = 6 and t = 2 using the exponential decay formula a = A * (1/2)^t. Explore the step-by-step process of using properties of exponents to determine the ratio and understand how the material decreases over time. Suitable for high school students in Grades 10-12.

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