Before decaying, a specific particle produced in a nuclear reactor...

The distance travelled by the particle in the laboratory frame is $L = 1 \text{ cm} = 10^{-2} \text{ m}$.

The speed of the particle is $v = 0.995c$.

The time lived by the particle in the laboratory frame is given by:

$\Delta t = \dfrac{L}{v} = \dfrac{10^{-2}}{0.995 \times 3 \times 10^8} = \dfrac{10^{-10}}{2.985} \approx 33.5 \times 10^{-12} \text{ s}$

The life of the particle in its own rest frame (proper time $\Delta t_0$) is related to the laboratory time by time dilation:

$\Delta t = \dfrac{\Delta t_0}{\sqrt{1 - \dfrac{v^2}{c^2}}}$

$\Rightarrow \Delta t_0 = \Delta t \sqrt{1 - \dfrac{v^2}{c^2}}$

Substituting the values:

$\Delta t_0 = 33.5 \times 10^{-12} \times \sqrt{1 - (0.995)^2}$

$\Delta t_0 = 33.5 \times 10^{-12} \times \sqrt{1 - 0.990025}$

$\Delta t_0 = 33.5 \times 10^{-12} \times \sqrt{0.009975}$

$\Delta t_0 \approx 33.5 \times 10^{-12} \times 0.09987$

$\Delta t_0 \approx 3.35 \times 10^{-12} \text{ s} = 3.35 \text{ ps}$