The exponential decay formula related to half-life is used to describe processes where the quantity of a substance decreases over time at a rate proportional to its current value. It is given by:

\[
N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}}
\]

Where:
- \(N(t)\) is the quantity of the substance at time \(t\),
- \(N_0\) is the initial quantity of the substance,
- \(T_{1/2}\) is the half-life, the time it takes for half of the substance to decay,
- \(t\) is the elapsed time.

This formula shows how the amount of a substance decreases by half every \(T_{1/2}\) time units.

### Breakdown:
1. **Initial quantity**: You start with \(N_0\).
2. **Time effect**: After each interval of the half-life \(T_{1/2}\), the amount is halved.
3. **Exponential decay**: The exponent \(\frac{t}{T_{1/2}}\) represents how many half-lives have passed.

Do you have any questions or need further details?

### Related Questions:
1. How do you calculate the half-life from a decay rate?
2. What is the difference between exponential decay and linear decay?
3. Can half-life be applied to things other than radioactive decay?
4. How does temperature affect the half-life of chemical reactions?
5. What is the significance of the decay constant in the half-life formula?

**Tip**: For radioactive decay, the half-life is constant and independent of the initial amount of the substance.

Explore the half-life exponential formula, which describes how the quantity of a substance decreases over time at a rate proportional to its current value. This problem covers exponential decay concepts and the half-life formula, suitable for Grades 10-12.

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