The decay constant of a radioactive substance for a and `beta` emission are `lambda_(a)` and `lambda_(beta)` respectively. It the substance emits a and `beta` simultaneously, the average half life of the material will be_______

Answer

To find the average half-life of a radioactive substance that emits both alpha (α) and beta (β) particles simultaneously, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Half-Life and Decay Constant**:
The half-life (T) of a radioactive substance is related to its decay constant (λ) by the formula:
\[ T = \frac{0.693}{\lambda} \]
where 0.693 is the natural logarithm of 2 (ln 2).

2. **Define the Half-Lives for Alpha and Beta Emissions**:
Let:
- \( T_{\alpha} \) be the half-life for alpha emission, given by:
\[ T_{\alpha} = \frac{0.693}{\lambda_{\alpha}} \]
- \( T_{\beta} \) be the half-life for beta emission, given by:
\[ T_{\beta} = \frac{0.693}{\lambda_{\beta}} \]

3. **Use the Formula for Combined Half-Life**:
When two decay processes occur simultaneously, the total decay constant \( \lambda_C \) for the combined process is the sum of the individual decay constants:
\[ \lambda_C = \lambda_{\alpha} + \lambda_{\beta} \]

4. **Calculate the Combined Half-Life**:
The half-life \( T_C \) for the combined emissions can be calculated using the decay constant \( \lambda_C \):
\[ T_C = \frac{0.693}{\lambda_C} = \frac{0.693}{\lambda_{\alpha} + \lambda_{\beta}} \]

5. **Final Expression for Average Half-Life**:
Thus, the average half-life of the material that emits both alpha and beta particles simultaneously is:
\[ T_C = \frac{0.693}{\lambda_{\alpha} + \lambda_{\beta}} \]

### Final Answer:
The average half-life of the material will be:
\[ T_C = \frac{0.693}{\lambda_{\alpha} + \lambda_{\beta}} \]