A radioactive sample disintegrates by 10% during one month. How much fraction will disintegrate in four months:

According to question, 10% of a radioactive samples disintegrates in one month.

Fraction of radioactive sample remain undecayed after one month, 0.9.
Let $N_0$ is initial number of samples.
Fraction of radioactive sample remain undecayed after four months is given as $\frac{\mathrm{N}}{{\mathrm{N}}_0}=(0.9{)}^4=0.6561$.

Fraction of sample decayed after four months

$=1-0.6561=0.3439\text{ or }34.69\mathrm{\%}\text{. }$